Code coverage tests

This page documents the degree to which the PARI/GP source code is tested by our public test suite, distributed with the source distribution in directory src/test/. This is measured by the gcov utility; we then process gcov output using the lcov frond-end.

We test a few variants depending on Configure flags on the pari.math.u-bordeaux.fr machine (x86_64 architecture), and agregate them in the final report:

The target is to exceed 90% coverage for all mathematical modules (given that branches depending on DEBUGLEVEL or DEBUGMEM are not covered). This script is run to produce the results below.

LCOV - code coverage report
Current view: top level - basemath - ifactor1.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.16.2 lcov report (development 29115-f22e516b23) Lines: 1555 1897 82.0 %
Date: 2024-02-22 08:05:55 Functions: 93 107 86.9 %
Legend: Lines: hit not hit

          Line data    Source code
       1             : /* Copyright (C) 2000  The PARI group.
       2             : 
       3             : This file is part of the PARI/GP package.
       4             : 
       5             : PARI/GP is free software; you can redistribute it and/or modify it under the
       6             : terms of the GNU General Public License as published by the Free Software
       7             : Foundation; either version 2 of the License, or (at your option) any later
       8             : version. It is distributed in the hope that it will be useful, but WITHOUT
       9             : ANY WARRANTY WHATSOEVER.
      10             : 
      11             : Check the License for details. You should have received a copy of it, along
      12             : with the package; see the file 'COPYING'. If not, write to the Free Software
      13             : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
      14             : #include "pari.h"
      15             : #include "paripriv.h"
      16             : 
      17             : #define DEBUGLEVEL DEBUGLEVEL_factorint
      18             : 
      19             : /***********************************************************************/
      20             : /**                       PRIMES IN SUCCESSION                        **/
      21             : /***********************************************************************/
      22             : 
      23             : /* map from prime residue classes mod 210 to their numbers in {0...47}.
      24             :  * Subscripts into this array take the form ((k-1)%210)/2, ranging from
      25             :  * 0 to 104.  Unused entries are */
      26             : #define NPRC 128 /* nonprime residue class */
      27             : 
      28             : static unsigned char prc210_no[] = {
      29             :   0, NPRC, NPRC, NPRC, NPRC, 1, 2, NPRC, 3, 4, NPRC, /* 21 */
      30             :   5, NPRC, NPRC, 6, 7, NPRC, NPRC, 8, NPRC, 9, /* 41 */
      31             :   10, NPRC, 11, NPRC, NPRC, 12, NPRC, NPRC, 13, 14, NPRC, /* 63 */
      32             :   NPRC, 15, NPRC, 16, 17, NPRC, NPRC, 18, NPRC, 19, /* 83 */
      33             :   NPRC, NPRC, 20, NPRC, NPRC, NPRC, 21, NPRC, 22, 23, NPRC, /* 105 */
      34             :   24, 25, NPRC, 26, NPRC, NPRC, NPRC, 27, NPRC, NPRC, /* 125 */
      35             :   28, NPRC, 29, NPRC, NPRC, 30, 31, NPRC, 32, NPRC, NPRC, /* 147 */
      36             :   33, 34, NPRC, NPRC, 35, NPRC, NPRC, 36, NPRC, 37, /* 167 */
      37             :   38, NPRC, 39, NPRC, NPRC, 40, 41, NPRC, NPRC, 42, NPRC, /* 189 */
      38             :   43, 44, NPRC, 45, 46, NPRC, NPRC, NPRC, NPRC, 47, /* 209 */
      39             : };
      40             : 
      41             : /* first differences of the preceding */
      42             : static unsigned char prc210_d1[] = {
      43             :   10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6,
      44             :   4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6,
      45             :   2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2,
      46             : };
      47             : 
      48             : static int
      49     1241446 : unextprime_overflow(ulong n)
      50             : {
      51             : #ifdef LONG_IS_64BIT
      52     1222026 :   return (n > (ulong)-59);
      53             : #else
      54       19420 :   return (n > (ulong)-5);
      55             : #endif
      56             : }
      57             : 
      58             : /* return 0 for overflow */
      59             : ulong
      60     1253336 : unextprime(ulong n)
      61             : {
      62             :   long rc, rc0, rcn;
      63             : 
      64     1253336 :   switch(n) {
      65        6718 :     case 0: case 1: case 2: return 2;
      66        2434 :     case 3: return 3;
      67        1668 :     case 4: case 5: return 5;
      68        1162 :     case 6: case 7: return 7;
      69             :   }
      70     1241354 :   if (unextprime_overflow(n)) return 0;
      71             :   /* here n > 7 */
      72     1241252 :   n |= 1; /* make it odd */
      73     1241252 :   rc = rc0 = n % 210;
      74             :   /* find next prime residue class mod 210 */
      75             :   for(;;)
      76             :   {
      77     2687008 :     rcn = (long)(prc210_no[rc>>1]);
      78     2687008 :     if (rcn != NPRC) break;
      79     1445756 :     rc += 2; /* cannot wrap since 209 is coprime and rc odd */
      80             :   }
      81     1241252 :   if (rc > rc0) n += rc - rc0;
      82             :   /* now find an actual (pseudo)prime */
      83             :   for(;;)
      84             :   {
      85    11549269 :     if (uisprime(n)) break;
      86    10308017 :     n += prc210_d1[rcn];
      87    10308017 :     if (++rcn > 47) rcn = 0;
      88             :   }
      89     1243881 :   return n;
      90             : }
      91             : 
      92             : GEN
      93      126839 : nextprime(GEN n)
      94             : {
      95             :   long rc, rc0, rcn;
      96      126839 :   pari_sp av = avma;
      97             : 
      98      126839 :   if (typ(n) != t_INT)
      99             :   {
     100          14 :     n = gceil(n);
     101          14 :     if (typ(n) != t_INT) pari_err_TYPE("nextprime",n);
     102             :   }
     103      126832 :   if (signe(n) <= 0) { set_avma(av); return gen_2; }
     104      126832 :   if (lgefint(n) == 3)
     105             :   {
     106      118696 :     ulong k = unextprime(uel(n,2));
     107      118696 :     set_avma(av);
     108      118696 :     if (k) return utoipos(k);
     109             : #ifdef LONG_IS_64BIT
     110           6 :     return uutoi(1,13);
     111             : #else
     112           1 :     return uutoi(1,15);
     113             : #endif
     114             :   }
     115             :   /* here n > 7 */
     116        8136 :   if (!mod2(n)) n = addui(1,n);
     117        8136 :   rc = rc0 = umodiu(n, 210);
     118             :   /* find next prime residue class mod 210 */
     119             :   for(;;)
     120             :   {
     121       17707 :     rcn = (long)(prc210_no[rc>>1]);
     122       17707 :     if (rcn != NPRC) break;
     123        9571 :     rc += 2; /* cannot wrap since 209 is coprime and rc odd */
     124             :   }
     125        8136 :   if (rc > rc0) n = addui(rc - rc0, n);
     126             :   /* now find an actual (pseudo)prime */
     127             :   for(;;)
     128             :   {
     129       84490 :     if (BPSW_psp(n)) break;
     130       76354 :     n = addui(prc210_d1[rcn], n);
     131       76354 :     if (++rcn > 47) rcn = 0;
     132             :   }
     133        8136 :   if (avma == av) return icopy(n);
     134        8136 :   return gerepileuptoint(av, n);
     135             : }
     136             : 
     137             : ulong
     138          32 : uprecprime(ulong n)
     139             : {
     140             :   long rc, rc0, rcn;
     141             :   { /* check if n <= 10 */
     142          32 :     if (n <= 1)  return 0;
     143          25 :     if (n == 2)  return 2;
     144          18 :     if (n <= 4)  return 3;
     145          18 :     if (n <= 6)  return 5;
     146          18 :     if (n <= 10) return 7;
     147             :   }
     148             :   /* here n >= 11 */
     149          18 :   if (!(n % 2)) n--;
     150          18 :   rc = rc0 = n % 210;
     151             :   /* find previous prime residue class mod 210 */
     152             :   for(;;)
     153             :   {
     154          36 :     rcn = (long)(prc210_no[rc>>1]);
     155          36 :     if (rcn != NPRC) break;
     156          18 :     rc -= 2; /* cannot wrap since 1 is coprime and rc odd */
     157             :   }
     158          18 :   if (rc < rc0) n += rc - rc0;
     159             :   /* now find an actual (pseudo)prime */
     160             :   for(;;)
     161             :   {
     162          36 :     if (uisprime(n)) break;
     163          18 :     if (--rcn < 0) rcn = 47;
     164          18 :     n -= prc210_d1[rcn];
     165             :   }
     166          18 :   return n;
     167             : }
     168             : 
     169             : GEN
     170          49 : precprime(GEN n)
     171             : {
     172             :   long rc, rc0, rcn;
     173          49 :   pari_sp av = avma;
     174             : 
     175          49 :   if (typ(n) != t_INT)
     176             :   {
     177          14 :     n = gfloor(n);
     178          14 :     if (typ(n) != t_INT) pari_err_TYPE("nextprime",n);
     179             :   }
     180          42 :   if (signe(n) <= 0) { set_avma(av); return gen_0; }
     181          42 :   if (lgefint(n) <= 3)
     182             :   {
     183          32 :     ulong k = uel(n,2);
     184          32 :     return gc_utoi(av, uprecprime(k));
     185             :   }
     186          10 :   if (!mod2(n)) n = subiu(n,1);
     187          10 :   rc = rc0 = umodiu(n, 210);
     188             :   /* find previous prime residue class mod 210 */
     189             :   for(;;)
     190             :   {
     191          20 :     rcn = (long)(prc210_no[rc>>1]);
     192          20 :     if (rcn != NPRC) break;
     193          10 :     rc -= 2; /* cannot wrap since 1 is coprime and rc odd */
     194             :   }
     195          10 :   if (rc0 > rc) n = subiu(n, rc0 - rc);
     196             :   /* now find an actual (pseudo)prime */
     197             :   for(;;)
     198             :   {
     199          48 :     if (BPSW_psp(n)) break;
     200          38 :     if (--rcn < 0) rcn = 47;
     201          38 :     n = subiu(n, prc210_d1[rcn]);
     202             :   }
     203          10 :   if (avma == av) return icopy(n);
     204          10 :   return gerepileuptoint(av, n);
     205             : }
     206             : 
     207             : /* Find next single-word prime strictly larger than p.
     208             :  * If **d is non-NULL (somewhere in a diffptr), this is p + *(*d)++;
     209             :  * otherwise imitate nextprime().
     210             :  * *rcn = NPRC or the correct residue class for the current p; we'll use this
     211             :  * to track the current prime residue class mod 210 once we're out of range of
     212             :  * the diffptr table, and we'll update it before that if it isn't NPRC.
     213             :  *
     214             :  * *q is incremented whenever q!=NULL and we wrap from 209 mod 210 to
     215             :  * 1 mod 210 */
     216             : ulong
     217     4634168 : snextpr(ulong p, byteptr *d, long *rcn, long *q, int (*ispsp)(ulong))
     218             : {
     219     4634168 :   if (**d)
     220             :   {
     221     4634168 :     byteptr dd = *d;
     222     4634168 :     long d1 = 0;
     223             : 
     224     4634168 :     NEXT_PRIME_VIADIFF(d1,dd);
     225             :     /* d1 = nextprime(p+1) - p */
     226     4634168 :     if (*rcn != NPRC)
     227             :     {
     228    16214716 :       while (d1 > 0)
     229             :       {
     230    11597848 :         d1 -= prc210_d1[*rcn];
     231    11597848 :         if (++*rcn > 47) { *rcn = 0; if (q) (*q)++; }
     232             :       }
     233             :       /* assert(d1 == 0) */
     234             :     }
     235     4634168 :     NEXT_PRIME_VIADIFF(p,*d);
     236     4634168 :     return p;
     237             :   }
     238           0 :   if (unextprime_overflow(p)) pari_err_OVERFLOW("snextpr");
     239             :   /* we are beyond the diffptr table, initialize if needed */
     240           0 :   if (*rcn == NPRC) *rcn = prc210_no[(p % 210) >> 1]; /* != NPRC */
     241             :   /* look for the next one */
     242             :   do {
     243           0 :     p += prc210_d1[*rcn];
     244           0 :     if (++*rcn > 47) { *rcn = 0; if (q) (*q)++; }
     245           0 :   } while (!ispsp(p));
     246           0 :   return p;
     247             : }
     248             : 
     249             : /********************************************************************/
     250             : /**                                                                **/
     251             : /**                     INTEGER FACTORIZATION                      **/
     252             : /**                                                                **/
     253             : /********************************************************************/
     254             : int factor_add_primes = 0, factor_proven = 0;
     255             : 
     256             : /***********************************************************************/
     257             : /**                                                                   **/
     258             : /**                 FACTORIZATION (ECM) -- GN Jul-Aug 1998            **/
     259             : /**   Integer factorization using the elliptic curves method (ECM).   **/
     260             : /**   ellfacteur() returns a non trivial factor of N, assuming N>0,   **/
     261             : /**   is composite, and has no prime divisor below 2^14 or so.        **/
     262             : /**   Thanks to Paul Zimmermann for much helpful advice and to        **/
     263             : /**   Guillaume Hanrot and Igor Schein for intensive testing          **/
     264             : /**                                                                   **/
     265             : /***********************************************************************/
     266             : #define nbcmax 64 /* max number of simultaneous curves */
     267             : 
     268             : static const ulong TB1[] = {
     269             :   142,172,208,252,305,370,450,545,661,801,972,1180,1430,
     270             :   1735,2100,2550,3090,3745,4540,5505,6675,8090,9810,11900,
     271             :   14420,17490,21200,25700,31160,37780UL,45810UL,55550UL,67350UL,
     272             :   81660UL,99010UL,120050UL,145550UL,176475UL,213970UL,259430UL,
     273             :   314550UL,381380UL,462415UL,560660UL,679780UL,824220UL,999340UL,
     274             :   1211670UL,1469110UL,1781250UL,2159700UL,2618600UL,3175000UL,
     275             :   3849600UL,4667500UL,5659200UL,6861600UL,8319500UL,10087100UL,
     276             :   12230300UL,14828900UL,17979600UL,21799700UL,26431500UL,
     277             :   32047300UL,38856400UL, /* 110 times that still fits into 32bits */
     278             : #ifdef LONG_IS_64BIT
     279             :   47112200UL,57122100UL,69258800UL,83974200UL,101816200UL,
     280             :   123449000UL,149678200UL,181480300UL,220039400UL,266791100UL,
     281             :   323476100UL,392204900UL,475536500UL,576573500UL,699077800UL,
     282             :   847610500UL,1027701900UL,1246057200UL,1510806400UL,1831806700UL,
     283             :   2221009800UL,2692906700UL,3265067200UL,3958794400UL,4799917500UL
     284             : #endif
     285             : };
     286             : static const ulong TB1_for_stage[] = {
     287             :  /* Start below the optimal B1 for finding factors which would just have been
     288             :   * missed by pollardbrent(), and escalate, changing curves to give good
     289             :   * coverage of the small factor ranges. Entries grow faster than what would
     290             :   * be optimal but a table instead of a 2D array keeps the code simple */
     291             :   500,520,560,620,700,800,900,1000,1150,1300,1450,1600,1800,2000,
     292             :   2200,2450,2700,2950,3250,3600,4000,4400,4850,5300,5800,6400,
     293             :   7100,7850,8700,9600,10600,11700,12900,14200,15700,17300,
     294             :   19000,21000,23200,25500,28000,31000,34500UL,38500UL,43000UL,
     295             :   48000UL,53800UL,60400UL,67750UL,76000UL,85300UL,95700UL,
     296             :   107400UL,120500UL,135400UL,152000UL,170800UL,191800UL,215400UL,
     297             :   241800UL,271400UL,304500UL,341500UL,383100UL,429700UL,481900UL,
     298             :   540400UL,606000UL,679500UL,761800UL,854100UL,957500UL,1073500UL
     299             : };
     300             : 
     301             : /* addition/doubling/multiplication of a point on an 'elliptic curve mod N'
     302             :  * may result in one of three things:
     303             :  * - a new bona fide point
     304             :  * - a point at infinity (denominator divisible by N)
     305             :  * - a point at infinity mod some p | N but finite mod q | N betraying itself
     306             :  *   by a denominator which has nontrivial gcd with N.
     307             :  *
     308             :  * In the second case, addition/doubling aborts, copying one of the summands
     309             :  * to the destination array of points unless they coincide.
     310             :  * Multiplication will stop at some unpredictable intermediate stage:  The
     311             :  * destination will contain _some_ multiple of the input point, but not
     312             :  * necessarily the desired one, which doesn't matter.  As long as we're
     313             :  * multiplying (B1 phase) we simply carry on with the next multiplier.
     314             :  * During the B2 phase, the only additions are the giant steps, and the
     315             :  * worst that can happen here is that we lose one residue class mod 210
     316             :  * of prime multipliers on 4 of the curves, so again, we ignore the problem
     317             :  * and just carry on.)
     318             :  *
     319             :  * Idea: select nbc curves mod N and one point P on each of them. For each
     320             :  * such P, compute [M]P = Q where M is the product of all powers <= B2 of
     321             :  * primes <= nextprime(B1). Then check whether [p]Q for p < nextprime(B2)
     322             :  * betrays a factor. This second stage looks separately at the primes in
     323             :  * each residue class mod 210, four curves at a time, and steps additively
     324             :  * to ever larger multipliers, by comparing X coordinates of points which we
     325             :  * would need to add in order to reach another prime multiplier in the same
     326             :  * residue class. 'Comparing' means that we accumulate a product of
     327             :  * differences of X coordinates, and from time to time take a gcd of this
     328             :  * product with N. Montgomery's multi-inverse trick is used heavily. */
     329             : 
     330             : /* *** auxiliary functions for ellfacteur: *** */
     331             : /* (Rx,Ry) <- (Px,Py)+(Qx,Qy) over Z/NZ, z=1/(Px-Qx). If Ry = NULL, don't set */
     332             : static void
     333     8397084 : FpE_add_i(GEN N, GEN z, GEN Px, GEN Py, GEN Qx, GEN Qy, GEN *Rx, GEN *Ry)
     334             : {
     335     8397084 :   GEN slope = modii(mulii(subii(Py, Qy), z), N);
     336     8397084 :   GEN t = subii(sqri(slope), addii(Qx, Px));
     337     8397084 :   affii(modii(t, N), *Rx);
     338     8397084 :   if (Ry) {
     339     8322620 :     t = subii(mulii(slope, subii(Px, *Rx)), Py);
     340     8322620 :     affii(modii(t, N), *Ry);
     341             :   }
     342     8397084 : }
     343             : /* X -> Z; cannot add on one of the curves: make sure Z contains
     344             :  * something useful before letting caller proceed */
     345             : static void
     346       28044 : ZV_aff(long n, GEN *X, GEN *Z)
     347             : {
     348       28044 :   if (X != Z) {
     349             :     long k;
     350     1601656 :     for (k = n; k--; ) affii(X[k],Z[k]);
     351             :   }
     352       28044 : }
     353             : 
     354             : /* Parallel addition on nbc curves, assigning the result to locations at and
     355             :  * following *X3, *Y3. (If Y-coords of result not desired, set Y=NULL.)
     356             :  * Safe even if (X3,Y3) = (X2,Y2), _not_ if (X1,Y1). It is also safe to
     357             :  * overwrite Y2 with X3. If nbc1 < nbc, the first summand is
     358             :  * assumed to hold only nbc1 distinct points, repeated as often as we need
     359             :  * them  (to add one point on each of a few curves to several other points on
     360             :  * the same curves): only used with nbc1 = nbc or nbc1 = 4 | nbc.
     361             :  *
     362             :  * Return 0 [SUCCESS], 1 [N | den], 2 [gcd(den, N) is a factor of N, preserved
     363             :  * in gl.
     364             :  * Stack space is bounded by a constant multiple of lgefint(N)*nbc:
     365             :  * - Phase 2 creates 12 items on the stack per iteration, of which 4 are twice
     366             :  *   as long and 1 is thrice as long as N, i.e. 18 units per iteration.
     367             :  * - Phase  1 creates 4 units.
     368             :  * Total can be as large as 4*nbcmax + 18*8 units; ecm_elladd2() is
     369             :  * just as bad, and elldouble() comes to 3*nbcmax + 29*8 units. */
     370             : static int
     371      252408 : ecm_elladd0(GEN N, GEN *gl, long nbc, long nbc1,
     372             :             GEN *X1, GEN *Y1, GEN *X2, GEN *Y2, GEN *X3, GEN *Y3)
     373             : {
     374      252408 :   const ulong mask = (nbc1 == 4)? 3: ~0UL; /*nbc1 = 4 or nbc*/
     375      252408 :   GEN W[2*nbcmax], *A = W+nbc; /* W[0],A[0] unused */
     376             :   long i;
     377      252408 :   pari_sp av = avma;
     378             : 
     379      252408 :   W[1] = subii(X1[0], X2[0]);
     380     7932408 :   for (i=1; i<nbc; i++)
     381             :   { /*prepare for multi-inverse*/
     382     7680000 :     A[i] = subii(X1[i&mask], X2[i]); /* don't waste time reducing mod N */
     383     7680000 :     W[i+1] = modii(mulii(A[i], W[i]), N);
     384             :   }
     385      252408 :   if (!invmod(W[nbc], N, gl))
     386             :   {
     387         368 :     if (!equalii(N,*gl)) return 2;
     388         343 :     ZV_aff(nbc, X2,X3);
     389         343 :     if (Y3) ZV_aff(nbc, Y2,Y3);
     390         343 :     return gc_int(av,1);
     391             :   }
     392             : 
     393     7926252 :   while (i--) /* nbc times */
     394             :   {
     395     7926252 :     pari_sp av2 = avma;
     396     7926252 :     GEN Px = X1[i&mask], Py = Y1[i&mask], Qx = X2[i], Qy = Y2[i];
     397     7926252 :     GEN z = i? mulii(*gl,W[i]): *gl; /*1/(Px-Qx)*/
     398     7926252 :     FpE_add_i(N,z,  Px,Py,Qx,Qy, X3+i, Y3? Y3+i: NULL);
     399     7926252 :     if (!i) break;
     400     7674212 :     set_avma(av2); *gl = modii(mulii(*gl, A[i]), N);
     401             :   }
     402      252040 :   return gc_int(av,0);
     403             : }
     404             : 
     405             : /* Shortcut, for use in cases where Y coordinates follow their corresponding
     406             :  * X coordinates, and first summand doesn't need to be repeated */
     407             : static int
     408      246226 : ecm_elladd(GEN N, GEN *gl, long nbc, GEN *X1, GEN *X2, GEN *X3) {
     409      246226 :   return ecm_elladd0(N, gl, nbc, nbc, X1, X1+nbc, X2, X2+nbc, X3, X3+nbc);
     410             : }
     411             : 
     412             : /* As ecm_elladd except it does twice as many additions (and hides even more
     413             :  * of the cost of the modular inverse); the net effect is the same as
     414             :  * ecm_elladd(nbc,X1,X2,X3) && ecm_elladd(nbc,X4,X5,X6). Safe to
     415             :  * have X2=X3, X5=X6, or X1,X2 coincide with X4,X5 in any order. */
     416             : static int
     417        7474 : ecm_elladd2(GEN N, GEN *gl, long nbc,
     418             :             GEN *X1, GEN *X2, GEN *X3, GEN *X4, GEN *X5, GEN *X6)
     419             : {
     420        7474 :   GEN *Y1 = X1+nbc, *Y2 = X2+nbc, *Y3 = X3+nbc;
     421        7474 :   GEN *Y4 = X4+nbc, *Y5 = X5+nbc, *Y6 = X6+nbc;
     422        7474 :   GEN W[4*nbcmax], *A = W+2*nbc; /* W[0],A[0] unused */
     423             :   long i, j;
     424        7474 :   pari_sp av = avma;
     425             : 
     426        7474 :   W[1] = subii(X1[0], X2[0]);
     427      235472 :   for (i=1; i<nbc; i++)
     428             :   {
     429      227998 :     A[i] = subii(X1[i], X2[i]); /* don't waste time reducing mod N here */
     430      227998 :     W[i+1] = modii(mulii(A[i], W[i]), N);
     431             :   }
     432      242946 :   for (j=0; j<nbc; i++,j++)
     433             :   {
     434      235472 :     A[i] = subii(X4[j], X5[j]);
     435      235472 :     W[i+1] = modii(mulii(A[i], W[i]), N);
     436             :   }
     437        7474 :   if (!invmod(W[2*nbc], N, gl))
     438             :   {
     439           7 :     if (!equalii(N,*gl)) return 2;
     440           7 :     ZV_aff(2*nbc, X2,X3); /* hack: 2*nbc => copy Y2->Y3 */
     441           7 :     ZV_aff(2*nbc, X5,X6); /* also copy Y5->Y6 */
     442           7 :     return gc_int(av,1);
     443             :   }
     444             : 
     445      242883 :   while (j--) /* nbc times */
     446             :   {
     447      235416 :     pari_sp av2 = avma;
     448      235416 :     GEN Px = X4[j], Py = Y4[j], Qx = X5[j], Qy = Y5[j];
     449      235416 :     GEN z = mulii(*gl,W[--i]); /*1/(Px-Qx)*/
     450      235416 :     FpE_add_i(N,z, Px,Py, Qx,Qy, X6+j,Y6+j);
     451      235416 :     set_avma(av2); *gl = modii(mulii(*gl, A[i]), N);
     452             :   }
     453      235416 :   while (i--) /* nbc times */
     454             :   {
     455      235416 :     pari_sp av2 = avma;
     456      235416 :     GEN Px = X1[i], Py = Y1[i], Qx = X2[i], Qy = Y2[i];
     457      235416 :     GEN z = i? mulii(*gl, W[i]): *gl; /*1/(Px-Qx)*/
     458      235416 :     FpE_add_i(N,z, Px,Py, Qx,Qy, X3+i,Y3+i);
     459      235416 :     if (!i) break;
     460      227949 :     set_avma(av2); *gl = modii(mulii(*gl, A[i]), N);
     461             :   }
     462        7467 :   return gc_int(av,0);
     463             : }
     464             : 
     465             : /* Parallel doubling on nbc curves, assigning the result to locations at
     466             :  * and following *X2.  Safe to be called with X2 equal to X1.  Return
     467             :  * value as for ecm_elladd.  If we find a point at infinity mod N,
     468             :  * and if X1 != X2, we copy the points at X1 to X2. */
     469             : static int
     470       42656 : elldouble(GEN N, GEN *gl, long nbc, GEN *X1, GEN *X2)
     471             : {
     472       42656 :   GEN *Y1 = X1+nbc, *Y2 = X2+nbc;
     473             :   GEN W[nbcmax+1]; /* W[0] unused */
     474             :   long i;
     475       42656 :   pari_sp av = avma;
     476       42656 :   /*W[0] = gen_1;*/ W[1] = Y1[0];
     477     1254208 :   for (i=1; i<nbc; i++) W[i+1] = modii(mulii(Y1[i], W[i]), N);
     478       42656 :   if (!invmod(W[nbc], N, gl))
     479             :   {
     480           0 :     if (!equalii(N,*gl)) return 2;
     481           0 :     ZV_aff(2*nbc,X1,X2); /* also copies Y1->Y2 */
     482           0 :     return gc_int(av,1);
     483             :   }
     484     1296864 :   while (i--) /* nbc times */
     485             :   {
     486             :     pari_sp av2;
     487     1254208 :     GEN v, w, L, z = i? mulii(*gl,W[i]): *gl;
     488     1254208 :     if (i) *gl = modii(mulii(*gl, Y1[i]), N);
     489     1254208 :     av2 = avma;
     490     1254208 :     L = modii(mulii(addui(1, mului(3, Fp_sqr(X1[i],N))), z), N);
     491     1254208 :     if (signe(L)) /* half of zero is still zero */
     492     1254208 :       L = shifti(mod2(L)? addii(L, N): L, -1);
     493     1254208 :     v = modii(subii(sqri(L), shifti(X1[i],1)), N);
     494     1254208 :     w = modii(subii(mulii(L, subii(X1[i], v)), Y1[i]), N);
     495     1254208 :     affii(v, X2[i]);
     496     1254208 :     affii(w, Y2[i]);
     497     1254208 :     set_avma(av2);
     498             :   }
     499       42656 :   return gc_int(av,0);
     500             : }
     501             : 
     502             : /* Parallel multiplication by an odd prime k on nbc curves, storing the
     503             :  * result to locations at and following *X2. Safe to be called with X2 = X1.
     504             :  * Return values as ecm_elladd. Uses (a simplified variant of) Montgomery's
     505             :  * PRAC algorithm; see ftp://ftp.cwi.nl/pub/pmontgom/Lucas.ps.gz .
     506             :  * With thanks to Paul Zimmermann for the reference.  --GN1998Aug13 */
     507             : static int
     508      218303 : get_rule(ulong d, ulong e)
     509             : {
     510      218303 :   if (d <= e + (e>>2)) /* floor(1.25*e) */
     511             :   {
     512       17351 :     if ((d+e)%3 == 0) return 0; /* rule 1 */
     513       10383 :     if ((d-e)%6 == 0) return 1;  /* rule 2 */
     514             :   }
     515             :   /* d <= 4*e but no ofl */
     516      211281 :   if ((d+3)>>2 <= e) return 2; /* rule 3, common case */
     517       12638 :   if ((d&1)==(e&1))  return 1; /* rule 4 = rule 2 */
     518        6561 :   if (!(d&1))        return 3; /* rule 5 */
     519        1839 :   if (d%3 == 0)      return 4; /* rule 6 */
     520         424 :   if ((d+e)%3 == 0)  return 5; /* rule 7 */
     521           0 :   if ((d-e)%3 == 0)  return 6; /* rule 8 */
     522             :   /* when we get here, e is even, otherwise one of rules 4,5 would apply */
     523           0 :   return 7; /* rule 9 */
     524             : }
     525             : 
     526             : /* PRAC implementation notes - main changes against the paper version:
     527             :  * (1) The general function [m+n]P = f([m]P,[n]P,[m-n]P) collapses (for m!=n)
     528             :  * to an ecm_elladd() which does not depend on the third argument; thus
     529             :  * references to the third variable (C in the paper) can be eliminated.
     530             :  * (2) Since our multipliers are prime, the outer loop of the paper
     531             :  * version executes only once, and thus is invisible above.
     532             :  * (3) The first step in the inner loop of the paper version will always be
     533             :  * rule 3, but the addition requested by this rule amounts to a doubling, and
     534             :  * will always be followed by a swap, so we have unrolled this first iteration.
     535             :  * (4) Simplifications in rules 6 and 7 are possible given the above, and we
     536             :  * save one addition in each of the two cases.  NB none of the other
     537             :  * ecm_elladd()s in the loop can ever degenerate into an elldouble.
     538             :  * (5) I tried to optimize for rule 3, which is used more frequently than all
     539             :  * others together, but it didn't improve things, so I removed the nested
     540             :  * tight loop again.  --GN */
     541             : /* The main loop body of ellfacteur() runs _slower_ under PRAC than under a
     542             :  * straightforward left-shift binary multiplication when N has <30 digits and
     543             :  * B1 is small;  PRAC wins when N and B1 get larger.  Weird. --GN */
     544             : /* k>2 assumed prime, XAUX = scratchpad */
     545             : static int
     546       27351 : ellmult(GEN N, GEN *gl, long nbc, ulong k, GEN *X1, GEN *X2, GEN *XAUX)
     547             : {
     548             :   ulong r, d, e, e1;
     549             :   int res;
     550       27351 :   GEN *A = X2, *B = XAUX, *T = XAUX + 2*nbc;
     551             : 
     552       27351 :   ZV_aff(2*nbc,X1,XAUX);
     553             :   /* first doubling picks up X1;  after this we'll be working in XAUX and
     554             :    * X2 only, mostly via A and B and T */
     555       27351 :   if ((res = elldouble(N, gl, nbc, X1, X2)) != 0) return res;
     556             : 
     557             :   /* split the work at the golden ratio */
     558       27351 :   r = (ulong)(k*0.61803398875 + .5);
     559       27351 :   d = k - r;
     560       27351 :   e = r - d; /* d+e == r, so no danger of ofl below */
     561      245507 :   while (d != e)
     562             :   { /* apply one of the nine transformations from PM's Table 4. */
     563      218303 :     switch(get_rule(d,e))
     564             :     {
     565        6968 :     case 0: /* rule 1 */
     566        6968 :       if ( (res = ecm_elladd(N, gl, nbc, A, B, T)) ) return res;
     567        6961 :       if ( (res = ecm_elladd2(N, gl, nbc, T, A, A, T, B, B)) != 0) return res;
     568        6954 :       e1 = d - e; d = (d + e1)/3; e = (e - e1)/3; break;
     569        6131 :     case 1: /* rules 2 and 4 */
     570        6131 :       if ( (res = ecm_elladd(N, gl, nbc, A, B, B)) ) return res;
     571        6117 :       if ( (res = elldouble(N, gl, nbc, A, A)) ) return res;
     572        6117 :       d = (d-e)>>1; break;
     573        4722 :     case 3: /* rule 5 */
     574        4722 :       if ( (res = elldouble(N, gl, nbc, A, A)) ) return res;
     575        4722 :       d >>= 1; break;
     576        1415 :     case 4: /* rule 6 */
     577        1415 :       if ( (res = elldouble(N, gl, nbc, A, T)) ) return res;
     578        1415 :       if ( (res = ecm_elladd(N, gl, nbc, T, A, A)) ) return res;
     579        1415 :       if ( (res = ecm_elladd(N, gl, nbc, A, B, B)) ) return res;
     580        1415 :       d = d/3 - e; break;
     581      198643 :     case 2: /* rule 3 */
     582      198643 :       if ( (res = ecm_elladd(N, gl, nbc, A, B, B)) ) return res;
     583      198524 :       d -= e; break;
     584         424 :     case 5: /* rule 7 */
     585         424 :       if ( (res = elldouble(N, gl, nbc, A, T)) ) return res;
     586         424 :       if ( (res = ecm_elladd2(N, gl, nbc, T, A, A, T, B, B)) != 0) return res;
     587         424 :       d = (d - 2*e)/3; break;
     588           0 :     case 6: /* rule 8 */
     589           0 :       if ( (res = ecm_elladd(N, gl, nbc, A, B, B)) ) return res;
     590           0 :       if ( (res = elldouble(N, gl, nbc, A, T)) ) return res;
     591           0 :       if ( (res = ecm_elladd(N, gl, nbc, T, A, A)) ) return res;
     592           0 :       d = (d - e)/3; break;
     593           0 :     case 7: /* rule 9 */
     594           0 :       if ( (res = elldouble(N, gl, nbc, B, B)) ) return res;
     595           0 :       e >>= 1; break;
     596             :     }
     597             :     /* swap d <-> e and A <-> B if necessary */
     598      218156 :     if (d < e) { lswap(d,e); pswap(A,B); }
     599             :   }
     600       27204 :   return ecm_elladd(N, gl, nbc, XAUX, X2, X2);
     601             : }
     602             : 
     603             : struct ECM {
     604             :   pari_timer T;
     605             :   long nbc, nbc2, seed;
     606             :   GEN *X, *XAUX, *XT, *XD, *XB, *XB2, *XH, *Xh, *Yh;
     607             : };
     608             : 
     609             : /* memory layout in ellfacteur():  a large array of GEN pointers, and one
     610             :  * huge chunk of memory containing all the actual GEN (t_INT) objects.
     611             :  * nbc is constant throughout the invocation:
     612             :  * - The B1 stage of each iteration through the main loop needs little
     613             :  * space:  enough for the X and Y coordinates of the current points,
     614             :  * and twice as much again as scratchpad for ellmult().
     615             :  * - The B2 stage, starting from some current set of points Q, needs, in
     616             :  * succession:
     617             :  *   + space for [2]Q, [4]Q, ..., [10]Q, and [p]Q for building the helix;
     618             :  *   + space for 48*nbc X and Y coordinates to hold the helix.  This could
     619             :  *   re-use [2]Q,...,[8]Q, but only with difficulty, since we don't
     620             :  *   know in advance which residue class mod 210 our p is going to be in.
     621             :  *   It can and should re-use [p]Q, though;
     622             :  *   + space for (temporarily [30]Q and then) [210]Q, [420]Q, and several
     623             :  *   further doublings until the giant step multiplier is reached.  This
     624             :  *   can re-use the remaining cells from above.  The computation of [210]Q
     625             :  *   will have been the last call to ellmult() within this iteration of the
     626             :  *   main loop, so the scratchpad is now also free to be re-used. We also
     627             :  *   compute [630]Q by a parallel addition;  we'll need it later to get the
     628             :  *   baby-step table bootstrapped a little faster.
     629             :  *   + Finally, for no more than 4 curves at a time, room for up to 1024 X
     630             :  *   coordinates only: the Y coordinates needed whilst setting up this baby
     631             :  *   step table are temporarily stored in the upper half, and overwritten
     632             :  *   during the last series of additions.
     633             :  *
     634             :  * Graphically:  after end of B1 stage (X,Y are the coords of Q):
     635             :  * +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--
     636             :  * | X Y |  scratch  | [2]Q| [4]Q| [6]Q| [8]Q|[10]Q|    ...    | ...
     637             :  * +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--
     638             :  * *X    *XAUX *XT   *XD                                       *XB
     639             :  *
     640             :  * [30]Q is computed from [10]Q.  [210]Q can go into XY, etc:
     641             :  * +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--
     642             :  * |[210]|[420]|[630]|[840]|[1680,3360,6720,...,2048*210]      |bstp table...
     643             :  * +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--
     644             :  * *X    *XAUX *XT   *XD      [*XG, somewhere here]            *XB .... *XH
     645             :  *
     646             :  * So we need (13 + 48) * 2 * nbc slots here + 4096 slots for the baby step
     647             :  * table (not all of which will be used when we start with a small B1, but
     648             :  * better to allocate and initialize ahead of time all the slots that might
     649             :  * be needed later).
     650             :  *
     651             :  * Note on memory locality:  During the B2 phase, accesses to the helix
     652             :  * (once it is set up) will be clustered by curves (4 out of nbc at a time).
     653             :  * Accesses to the baby steps table will wander from one end of the array to
     654             :  * the other and back, one such cycle per giant step, and during a full cycle
     655             :  * we would expect on the order of 2E4 accesses when using the largest giant
     656             :  * step size.  Thus we shouldn't be doing too bad with respect to thrashing
     657             :  * a 512KBy L2 cache.  However, we don't want the baby step table to grow
     658             :  * larger than this, even if it would reduce the number of EC operations by a
     659             :  * few more per cent for very large B2, lest cache thrashing slow down
     660             :  * everything disproportionally. --GN */
     661             : /* Auxiliary routines need < (3*nbc+240)*lN words on the PARI stack, in
     662             :  * addition to the spc*(lN+1) words occupied by our main table. */
     663             : static void
     664          71 : ECM_alloc(struct ECM *E, long lN)
     665             : {
     666          71 :   const long bstpmax = 1024; /* max number of baby step table entries */
     667          71 :   long spc = (13 + 48) * E->nbc2 + bstpmax * 4;
     668          71 :   long len = spc + 385 + spc*lN;
     669          71 :   long i, tw = _evallg(lN) | evaltyp(t_INT);
     670          71 :   GEN w, *X = (GEN*)new_chunk(len);
     671             :   /* hack for X[i] = cgeti(lN). X = current point in B1 phase */
     672          71 :   w = (GEN)(X + spc + 385);
     673      448023 :   for (i = spc-1; i >= 0; i--) { X[i] = w; *w = tw; w += lN; }
     674          71 :   E->X = X;
     675          71 :   E->XAUX = E->X    + E->nbc2; /* scratchpad for ellmult() */
     676          71 :   E->XT   = E->XAUX + E->nbc2; /* ditto, will later hold [3*210]Q */
     677          71 :   E->XD   = E->XT   + E->nbc2; /* room for various multiples */
     678          71 :   E->XB   = E->XD   + 10*E->nbc2; /* start of baby steps table */
     679          71 :   E->XB2  = E->XB   + 2 * bstpmax; /* middle of baby steps table */
     680          71 :   E->XH   = E->XB2  + 2 * bstpmax; /* end of bstps table, start of helix */
     681          71 :   E->Xh   = E->XH   + 48*E->nbc2; /* little helix, X coords */
     682          71 :   E->Yh   = E->XH   + 192;     /* ditto, Y coords */
     683             :   /* XG,YG set inside the main loop, since they depend on B2 */
     684             :   /* E.Xh range of 384 pointers not set; these will later duplicate the pointers
     685             :    * in the E.XH range, 4 curves at a time. Some of the cells reserved here for
     686             :    * the E.XB range will never be used, instead, we'll warp the pointers to
     687             :    * connect to (read-only) GENs in the X/E.XD range */
     688          71 : }
     689             : /* N.B. E->seed is not initialized here */
     690             : static void
     691          71 : ECM_init(struct ECM *E, GEN N, long nbc)
     692             : {
     693          71 :   if (nbc < 0)
     694             :   { /* choose a sensible default */
     695          64 :     const long size = expi(N) + 1;
     696          64 :     nbc = ((size >> 3) << 2) - 80;
     697          64 :     if (nbc < 8) nbc = 8;
     698             :   }
     699          71 :   if (nbc > nbcmax) nbc = nbcmax;
     700          71 :   E->nbc = nbc;
     701          71 :   E->nbc2 = nbc << 1;
     702          71 :   ECM_alloc(E, lgefint(N));
     703          71 : }
     704             : 
     705             : static GEN
     706         114 : ECM_loop(struct ECM *E, GEN N, ulong B1)
     707             : {
     708         114 :   const ulong B2 = 110 * B1, B2_rt = usqrt(B2);
     709         114 :   const ulong nbc = E->nbc, nbc2 = E->nbc2;
     710             :   pari_sp av1, avtmp;
     711         114 :   byteptr d0, d = diffptr;
     712             :   long i, gse, gss, bstp, bstp0, rcn0, rcn;
     713             :   ulong B2_p, m, p, p0;
     714             :   GEN g, *XG, *YG;
     715         114 :   GEN *X = E->X, *XAUX = E->XAUX, *XT = E->XT, *XD = E->XD;
     716         114 :   GEN *XB = E->XB, *XB2 = E->XB2, *XH = E->XH, *Xh = E->Xh, *Yh = E->Yh;
     717             :   /* pick curves */
     718        4818 :   for (i = nbc2; i--; ) affui(E->seed++, X[i]);
     719             :   /* pick giant step exponent and size */
     720         114 :   gse = B1 < 656
     721             :           ? (B1 < 200? 5: 6)
     722         114 :           : (B1 < 10500
     723             :             ? (B1 < 2625? 7: 8)
     724             :             : (B1 < 42000? 9: 10));
     725         114 :   gss = 1UL << gse;
     726             :   /* With 32 baby steps, a giant step corresponds to 32*420 = 13440,
     727             :    * appropriate for the smallest B2s. With 1024, a giant step will be 430080;
     728             :    * appropriate for B1 >~ 42000, where 512 baby steps would imply roughly
     729             :    * the same number of curve additions. */
     730         114 :   XG = XT + gse*nbc2; /* will later hold [2^(gse+1)*210]Q */
     731         114 :   YG = XG + nbc;
     732             : 
     733         114 :   if (DEBUGLEVEL >= 4) {
     734           0 :     err_printf("ECM: time = %6ld ms\nECM: B1 = %4lu,", timer_delay(&E->T), B1);
     735           0 :     err_printf("\tB2 = %6lu,\tgss = %4ld*420\n", B2, gss);
     736             :   }
     737         114 :   p = 0;
     738         114 :   NEXT_PRIME_VIADIFF(p,d);
     739             : 
     740             :   /* ---B1 PHASE--- */
     741             :   /* treat p=2 separately */
     742         114 :   B2_p = B2 >> 1;
     743        1911 :   for (m=1; m<=B2_p; m<<=1)
     744             :   {
     745        1797 :     int fl = elldouble(N, &g, nbc, X, X);
     746        1797 :     if (fl > 1) return g; else if (fl) break;
     747             :   }
     748         114 :   rcn = NPRC; /* multipliers begin at the beginning */
     749             :   /* p=3,...,nextprime(B1) */
     750        7133 :   while (p < B1 && p <= B2_rt)
     751             :   {
     752        7026 :     pari_sp av2 = avma;
     753        7026 :     p = snextpr(p, &d, &rcn, NULL, uisprime);
     754        7026 :     B2_p = B2/p; /* beware integer overflow on 32-bit CPUs */
     755       23792 :     for (m=1; m<=B2_p; m*=p)
     756             :     {
     757       16899 :       int fl = ellmult(N, &g, nbc, p, X, X, XAUX);
     758       16899 :       if (fl > 1) return g; else if (fl) break;
     759       16766 :       set_avma(av2);
     760             :     }
     761        7019 :     set_avma(av2);
     762             :   }
     763             :   /* primes p larger than sqrt(B2) appear only to the 1st power */
     764       10274 :   while (p < B1)
     765             :   {
     766       10185 :     pari_sp av2 = avma;
     767       10185 :     p = snextpr(p, &d, &rcn, NULL, uisprime);
     768       10185 :     if (ellmult(N, &g, nbc, p, X, X, XAUX) > 1) return g;
     769       10167 :     set_avma(av2);
     770             :   }
     771          89 :   if (DEBUGLEVEL >= 4) {
     772           0 :     err_printf("ECM: time = %6ld ms, B1 phase done, ", timer_delay(&E->T));
     773           0 :     err_printf("p = %lu, setting up for B2\n", p);
     774             :   }
     775             : 
     776             :   /* ---B2 PHASE--- */
     777             :   /* compute [2]Q,...,[10]Q, needed to build the helix */
     778          89 :   if (elldouble(N, &g, nbc, X, XD) > 1) return g; /*[2]Q*/
     779          89 :   if (elldouble(N, &g, nbc, XD, XD + nbc2) > 1) return g; /*[4]Q*/
     780          89 :   if (ecm_elladd(N, &g, nbc,
     781          89 :         XD, XD + nbc2, XD + (nbc<<2)) > 1) return g; /* [6]Q */
     782          89 :   if (ecm_elladd2(N, &g, nbc,
     783             :         XD, XD + (nbc<<2), XT + (nbc<<3),
     784          89 :         XD + nbc2, XD + (nbc<<2), XD + (nbc<<3)) > 1)
     785           0 :     return g; /* [8]Q and [10]Q */
     786          89 :   if (DEBUGLEVEL >= 7) err_printf("\t(got [2]Q...[10]Q)\n");
     787             : 
     788             :   /* get next prime (still using the foolproof test) */
     789          89 :   p = snextpr(p, &d, &rcn, NULL, uisprime);
     790             :   /* make sure we have the residue class number (mod 210) */
     791          89 :   if (rcn == NPRC)
     792             :   {
     793          89 :     rcn = prc210_no[(p % 210) >> 1];
     794          89 :     if (rcn == NPRC)
     795             :     {
     796           0 :       err_printf("ECM: %lu should have been prime but isn\'t\n", p);
     797           0 :       pari_err_BUG("ellfacteur");
     798             :     }
     799             :   }
     800             : 
     801             :   /* compute [p]Q and put it into its place in the helix */
     802          89 :   if (ellmult(N, &g, nbc, p, X, XH + rcn*nbc2, XAUX) > 1)
     803           0 :     return g;
     804          89 :   if (DEBUGLEVEL >= 7)
     805           0 :     err_printf("\t(got [p]Q, p = %lu = prc210_rp[%ld] mod 210)\n", p, rcn);
     806             : 
     807             :   /* save current p, d, and rcn;  we'll need them more than once below */
     808          89 :   p0 = p;
     809          89 :   d0 = d;
     810          89 :   rcn0 = rcn; /* remember where the helix wraps */
     811          89 :   bstp0 = 0; /* p is at baby-step offset 0 from itself */
     812             : 
     813             :   /* fill up the helix, stepping forward through the prime residue classes
     814             :    * mod 210 until we're back at the r'class of p0.  Keep updating p so
     815             :    * that we can print meaningful diagnostics if a factor shows up; don't
     816             :    * bother checking which of these p's are in fact prime */
     817        4272 :   for (i = 47; i; i--) /* 47 iterations */
     818             :   {
     819        4183 :     ulong dp = (ulong)prc210_d1[rcn];
     820        4183 :     p += dp;
     821        4183 :     if (rcn == 47)
     822             :     { /* wrap mod 210 */
     823          89 :       if (ecm_elladd(N, &g, nbc, XT+dp*nbc, XH+rcn*nbc2, XH) > 1) return g;
     824          89 :       rcn = 0; continue;
     825             :     }
     826        4094 :     if (ecm_elladd(N, &g, nbc, XT+dp*nbc, XH+rcn*nbc2, XH+rcn*nbc2+nbc2) > 1)
     827           0 :       return g;
     828        4094 :     rcn++;
     829             :   }
     830          89 :   if (DEBUGLEVEL >= 7) err_printf("\t(got initial helix)\n");
     831             :   /* compute [210]Q etc, needed for the baby step table */
     832          89 :   if (ellmult(N, &g, nbc, 3, XD + (nbc<<3), X, XAUX) > 1) return g;
     833          89 :   if (ellmult(N, &g, nbc, 7, X, X, XAUX) > 1) return g; /* [210]Q */
     834             :   /* this was the last call to ellmult() in the main loop body; may now
     835             :    * overwrite XAUX and slots XD and following */
     836          89 :   if (elldouble(N, &g, nbc, X, XAUX) > 1) return g; /* [420]Q */
     837          89 :   if (ecm_elladd(N, &g, nbc, X, XAUX, XT) > 1) return g;/*[630]Q*/
     838          89 :   if (ecm_elladd(N, &g, nbc, X, XT, XD) > 1) return g;  /*[840]Q*/
     839         652 :   for (i=1; i <= gse; i++)
     840         563 :     if (elldouble(N, &g, nbc, XT + i*nbc2, XD + i*nbc2) > 1) return g;
     841             :   /* (the last iteration has initialized XG to [210*2^(gse+1)]Q) */
     842             : 
     843          89 :   if (DEBUGLEVEL >= 4)
     844           0 :     err_printf("ECM: time = %6ld ms, entering B2 phase, p = %lu\n",
     845             :                timer_delay(&E->T), p);
     846             : 
     847         433 :   for (i = nbc - 4; i >= 0; i -= 4)
     848             :   { /* loop over small sets of 4 curves at a time */
     849             :     GEN *Xb;
     850             :     long j, k;
     851         351 :     if (DEBUGLEVEL >= 6)
     852           0 :       err_printf("ECM: finishing curves %ld...%ld\n", i, i+3);
     853             :     /* Copy relevant pointers from XH to Xh. Memory layout in XH:
     854             :      * nbc X coordinates, nbc Y coordinates for residue class
     855             :      * 1 mod 210, then the same for r.c. 11 mod 210, etc. Memory layout for
     856             :      * Xh is: four X coords for 1 mod 210, four for 11 mod 210, ..., four
     857             :      * for 209 mod 210, then the corresponding Y coordinates in the same
     858             :      * order. This allows a giant step on Xh using just three calls to
     859             :      * ecm_elladd0() each acting on 64 points in parallel */
     860       17199 :     for (j = 48; j--; )
     861             :     {
     862       16848 :       k = nbc2*j + i;
     863       16848 :       m = j << 2; /* X coordinates */
     864       16848 :       Xh[m]   = XH[k];   Xh[m+1] = XH[k+1];
     865       16848 :       Xh[m+2] = XH[k+2]; Xh[m+3] = XH[k+3];
     866       16848 :       k += nbc; /* Y coordinates */
     867       16848 :       Yh[m]   = XH[k];   Yh[m+1] = XH[k+1];
     868       16848 :       Yh[m+2] = XH[k+2]; Yh[m+3] = XH[k+3];
     869             :     }
     870             :     /* Build baby step table of X coords of multiples of [210]Q.  XB[4*j]
     871             :      * will point at X coords on four curves from [(j+1)*210]Q.  Until
     872             :      * we're done, we need some Y coords as well, which we keep in the
     873             :      * second half of the table, overwriting them at the end when gse=10.
     874             :      * Multiples which we already have  (by 1,2,3,4,8,16,...,2^gse) are
     875             :      * entered simply by copying the pointers, ignoring the few slots in w
     876             :      * that were initially reserved for them. Here are the initial entries */
     877        1053 :     for (Xb=XB,k=2,j=i; k--; Xb=XB2,j+=nbc) /* first X, then Y coords */
     878             :     {
     879         702 :       Xb[0]  = X[j];      Xb[1]  = X[j+1]; /* [210]Q */
     880         702 :       Xb[2]  = X[j+2];    Xb[3]  = X[j+3];
     881         702 :       Xb[4]  = XAUX[j];   Xb[5]  = XAUX[j+1]; /* [420]Q */
     882         702 :       Xb[6]  = XAUX[j+2]; Xb[7]  = XAUX[j+3];
     883         702 :       Xb[8]  = XT[j];     Xb[9]  = XT[j+1]; /* [630]Q */
     884         702 :       Xb[10] = XT[j+2];   Xb[11] = XT[j+3];
     885         702 :       Xb += 4; /* points at [420]Q */
     886             :       /* ... entries at powers of 2 times 210 .... */
     887        4337 :       for (m = 2; m < (ulong)gse+k; m++) /* omit Y coords of [2^gse*210]Q */
     888             :       {
     889        3635 :         long m2 = m*nbc2 + j;
     890        3635 :         Xb += (2UL<<m); /* points at [2^m*210]Q */
     891        3635 :         Xb[0] = XAUX[m2];   Xb[1] = XAUX[m2+1];
     892        3635 :         Xb[2] = XAUX[m2+2]; Xb[3] = XAUX[m2+3];
     893             :       }
     894             :     }
     895         351 :     if (DEBUGLEVEL >= 7)
     896           0 :       err_printf("\t(extracted precomputed helix / baby step entries)\n");
     897             :     /* ... glue in between, up to 16*210 ... */
     898         351 :     if (ecm_elladd0(N, &g, 12, 4, /* 12 pts + (4 pts replicated thrice) */
     899             :           XB + 12, XB2 + 12,
     900             :           XB,      XB2,
     901           0 :           XB + 16, XB2 + 16) > 1) return g; /*4+{1,2,3} = {5,6,7}*/
     902         351 :     if (ecm_elladd0(N, &g, 28, 4, /* 28 pts + (4 pts replicated 7fold) */
     903             :           XB + 28, XB2 + 28,
     904             :           XB,      XB2,
     905           0 :           XB + 32, XB2 + 32) > 1) return g;/*8+{1...7} = {9...15}*/
     906             :     /* ... and the remainder of the lot */
     907        1291 :     for (m = 5; m <= (ulong)gse; m++)
     908             :     { /* fill in from 2^(m-1)+1 to 2^m-1 in chunks of 64 and 60 points */
     909         940 :       ulong m2 = 2UL << m; /* will point at 2^(m-1)+1 */
     910        2033 :       for (j = 0; (ulong)j < m2-64; j+=64) /* executed 0 times when m = 5 */
     911             :       {
     912        1934 :         if (ecm_elladd0(N, &g, 64, 4,
     913        1093 :               XB + m2-4, XB2 + m2-4,
     914        1093 :               XB + j,    XB2 + j,
     915        1934 :               XB + m2+j, (m<(ulong)gse? XB2+m2+j: NULL)) > 1)
     916           0 :           return g;
     917             :       } /* j = m2-64 here, 60 points left */
     918        1291 :       if (ecm_elladd0(N, &g, 60, 4,
     919         940 :             XB + m2-4, XB2 + m2-4,
     920         940 :             XB + j,    XB2 + j,
     921        1291 :             XB + m2+j, (m<(ulong)gse? XB2+m2+j: NULL)) > 1)
     922           0 :         return g;
     923             :       /* when m=gse, drop Y coords of result, and when both equal 1024,
     924             :        * overwrite Y coords of second argument with X coords of result */
     925             :     }
     926         351 :     if (DEBUGLEVEL >= 7) err_printf("\t(baby step table complete)\n");
     927             :     /* initialize a few other things */
     928         351 :     bstp = bstp0;
     929         351 :     p = p0; d = d0; rcn = rcn0;
     930         351 :     g = gen_1; av1 = avma;
     931             :     /* scratchspace for prod (x_i-x_j) */
     932         351 :     avtmp = (pari_sp)new_chunk(8 * lgefint(N));
     933             :     /* The correct entry in XB to use depends on bstp and on where we are
     934             :      * on the helix. As we skip from prime to prime, bstp is incremented
     935             :      * by snextpr each time we wrap around through residue class number 0
     936             :      * (1 mod 210), but the baby step should not be taken until rcn>=rcn0,
     937             :      * i.e. until we pass again the residue class of p0.
     938             :      *
     939             :      * The correct signed multiplier is thus k = bstp - (rcn < rcn0),
     940             :      * and the offset from XB is four times (|k| - 1).  When k=0, we ignore
     941             :      * the current prime: if it had led to a factorization, this
     942             :      * would have been noted during the last giant step, or -- when we
     943             :      * first get here -- whilst initializing the helix.  When k > gss,
     944             :      * we must do a giant step and bump bstp back by -2*gss.
     945             :      *
     946             :      * The gcd of the product of X coord differences against N is taken just
     947             :      * before we do a giant step. */
     948     4617212 :     while (p < B2)
     949             :     {/* loop over probable primes p0 < p <= nextprime(B2), inserting giant
     950             :       * steps as necessary */
     951     4616868 :       p = snextpr(p, &d, &rcn, &bstp, uis2psp); /* next probable prime */
     952             :       /* work out the corresponding baby-step multiplier */
     953     4616868 :       k = bstp - (rcn < rcn0 ? 1 : 0);
     954     4616868 :       if (k > gss)
     955             :       { /* giant-step time, take gcd */
     956        1156 :         g = gcdii(g, N);
     957        1156 :         if (!is_pm1(g) && !equalii(g, N)) return g;
     958        1149 :         g = gen_1; set_avma(av1);
     959        2298 :         while (k > gss)
     960             :         { /* giant step */
     961        1149 :           if (DEBUGLEVEL >= 7) err_printf("\t(giant step at p = %lu)\n", p);
     962        1149 :           if (ecm_elladd0(N, &g, 64, 4, XG + i, YG + i,
     963           0 :                 Xh, Yh, Xh, Yh) > 1) return g;
     964        1149 :           if (ecm_elladd0(N, &g, 64, 4, XG + i, YG + i,
     965             :                 Xh + 64, Yh + 64, Xh + 64, Yh + 64) > 1)
     966           0 :             return g;
     967        1149 :           if (ecm_elladd0(N, &g, 64, 4, XG + i, YG + i,
     968             :                 Xh + 128, Yh + 128, Xh + 128, Yh + 128) > 1)
     969           0 :             return g;
     970        1149 :           bstp -= (gss << 1);
     971        1149 :           k = bstp - (rcn < rcn0? 1: 0); /* recompute multiplier */
     972             :         }
     973             :       }
     974     4616861 :       if (!k) continue; /* point of interest is already in Xh */
     975     4590320 :       if (k < 0) k = -k;
     976     4590320 :       m = ((ulong)k - 1) << 2;
     977             :       /* accumulate product of differences of X coordinates */
     978     4590320 :       j = rcn<<2;
     979     4590320 :       avma = avtmp; /* go to garbage zone; don't use set_avma */
     980     4590320 :       g = modii(mulii(g, subii(XB[m],   Xh[j])), N);
     981     4590320 :       g = modii(mulii(g, subii(XB[m+1], Xh[j+1])), N);
     982     4590320 :       g = modii(mulii(g, subii(XB[m+2], Xh[j+2])), N);
     983     4590320 :       g = mulii(g, subii(XB[m+3], Xh[j+3]));
     984     4590320 :       set_avma(av1);
     985     4590320 :       g = modii(g, N);
     986             :     }
     987         344 :     set_avma(av1);
     988             :   }
     989          82 :   return NULL;
     990             : }
     991             : 
     992             : /* ellfacteur() tuned to be useful as a first stage before MPQS, especially for
     993             :  * large arguments, when 'insist' is false, and now also for the case when
     994             :  * 'insist' is true, vaguely following suggestions by Paul Zimmermann
     995             :  * (http://www.loria.fr/~zimmerma/records/ecmnet.html). --GN 1998Jul,Aug */
     996             : static GEN
     997         946 : ellfacteur(GEN N, int insist)
     998             : {
     999         946 :   const long size = expi(N) + 1;
    1000         946 :   pari_sp av = avma;
    1001             :   struct ECM E;
    1002         946 :   long nbc, dsn, dsnmax, rep = 0;
    1003         946 :   if (insist)
    1004             :   {
    1005           7 :     const long DSNMAX = numberof(TB1)-1;
    1006           7 :     dsnmax = (size >> 2) - 10;
    1007           7 :     if (dsnmax < 0) dsnmax = 0;
    1008           0 :     else if (dsnmax > DSNMAX) dsnmax = DSNMAX;
    1009           7 :     E.seed = 1 + (nbcmax<<7)*(size&0xffff); /* seed for choice of curves */
    1010             : 
    1011           7 :     dsn = (size >> 3) - 5;
    1012           7 :     if (dsn < 0) dsn = 0; else if (dsn > 47) dsn = 47;
    1013             :     /* pick up the torch where noninsistent stage would have given up */
    1014           7 :     nbc = dsn + (dsn >> 2) + 9; /* 8 or more curves in parallel */
    1015           7 :     nbc &= ~3; /* 4 | nbc */
    1016             :   }
    1017             :   else
    1018             :   {
    1019         939 :     dsn = (size - 140) >> 3;
    1020         939 :     if (dsn < 0)
    1021             :     {
    1022             : #ifndef __EMX__ /* unless DOS/EMX: MPQS's disk access is abysmally slow */
    1023         875 :       if (DEBUGLEVEL >= 4)
    1024           0 :         err_printf("ECM: number too small to justify this stage\n");
    1025         875 :       return NULL; /* too small, decline the task */
    1026             : #endif
    1027             :       dsn = 0;
    1028          64 :     } else if (dsn > 12) dsn = 12;
    1029          64 :     rep = (size <= 248 ?
    1030          64 :            (size <= 176 ? (size - 124) >> 4 : (size - 148) >> 3) :
    1031          18 :            (size - 224) >> 1);
    1032             : #ifdef __EMX__ /* DOS/EMX: extra rounds (shun MPQS) */
    1033             :     rep += 20;
    1034             : #endif
    1035          64 :     dsnmax = 72;
    1036             :     /* Use disjoint sets of curves for non-insist and insist phases; moreover,
    1037             :      * repeated calls acting on factors of the same original number should try
    1038             :      * to use fresh curves. The following achieves this */
    1039          64 :     E.seed = 1 + (nbcmax<<3)*(size & 0xf);
    1040          64 :     nbc = -1;
    1041             :   }
    1042          71 :   ECM_init(&E, N, nbc);
    1043          71 :   if (DEBUGLEVEL >= 4)
    1044             :   {
    1045           0 :     timer_start(&E.T);
    1046           0 :     err_printf("ECM: working on %ld curves at a time; initializing", E.nbc);
    1047           0 :     if (!insist)
    1048             :     {
    1049           0 :       if (rep == 1) err_printf(" for one round");
    1050           0 :       else          err_printf(" for up to %ld rounds", rep);
    1051             :     }
    1052           0 :     err_printf("...\n");
    1053             :   }
    1054          71 :   if (dsn > dsnmax) dsn = dsnmax;
    1055             :   for(;;)
    1056          43 :   {
    1057         114 :     ulong B1 = insist? TB1[dsn]: TB1_for_stage[dsn];
    1058         114 :     GEN g = ECM_loop(&E, N, B1);
    1059         114 :     if (g)
    1060             :     {
    1061          32 :       if (DEBUGLEVEL >= 4)
    1062           0 :         err_printf("ECM: time = %6ld ms\n\tfound factor = %Ps\n",
    1063             :                    timer_delay(&E.T), g);
    1064          32 :       return gerepilecopy(av, g);
    1065             :     }
    1066          82 :     if (dsn < dsnmax)
    1067             :     {
    1068          75 :       if (insist) dsn++;
    1069          75 :       else { dsn += 2; if (dsn > dsnmax) dsn = dsnmax; }
    1070             :     }
    1071          82 :     if (!insist && !--rep)
    1072             :     {
    1073          39 :       if (DEBUGLEVEL >= 4)
    1074           0 :         err_printf("ECM: time = %6ld ms,\tellfacteur giving up.\n",
    1075             :                    timer_delay(&E.T));
    1076          39 :       return gc_NULL(av);
    1077             :     }
    1078             :   }
    1079             : }
    1080             : /* assume rounds >= 1, seed >= 1, B1 <= ULONG_MAX / 110 */
    1081             : GEN
    1082           0 : Z_ECM(GEN N, long rounds, long seed, ulong B1)
    1083             : {
    1084           0 :   pari_sp av = avma;
    1085             :   struct ECM E;
    1086             :   long i;
    1087           0 :   E.seed = seed;
    1088           0 :   ECM_init(&E, N, -1);
    1089           0 :   if (DEBUGLEVEL >= 4) timer_start(&E.T);
    1090           0 :   for (i = rounds; i--; )
    1091             :   {
    1092           0 :     GEN g = ECM_loop(&E, N, B1);
    1093           0 :     if (g) return gerepilecopy(av, g);
    1094             :   }
    1095           0 :   return gc_NULL(av);
    1096             : }
    1097             : 
    1098             : /***********************************************************************/
    1099             : /**                                                                   **/
    1100             : /**                FACTORIZATION (Pollard-Brent rho) --GN1998Jun18-26 **/
    1101             : /**  pollardbrent() returns a nontrivial factor of n, assuming n is   **/
    1102             : /**  composite and has no small prime divisor, or NULL if going on    **/
    1103             : /**  would take more time than we want to spend.  Sometimes it finds  **/
    1104             : /**  more than one factor, and returns a structure suitable for       **/
    1105             : /**  interpretation by ifac_crack. (Cf Algo 8.5.2 in ACiCNT)          **/
    1106             : /**                                                                   **/
    1107             : /***********************************************************************/
    1108             : #define VALUE(x) gel(x,0)
    1109             : #define EXPON(x) gel(x,1)
    1110             : #define CLASS(x) gel(x,2)
    1111             : 
    1112             : INLINE void
    1113       50563 : INIT(GEN x, GEN v, GEN e, GEN c) {
    1114       50563 :   VALUE(x) = v;
    1115       50563 :   EXPON(x) = e;
    1116       50563 :   CLASS(x) = c;
    1117       50563 : }
    1118             : static void
    1119       44652 : ifac_delete(GEN x) { INIT(x,NULL,NULL,NULL); }
    1120             : 
    1121             : static void
    1122           0 : rho_dbg(pari_timer *T, long c, long msg_mask)
    1123             : {
    1124           0 :   if (c & msg_mask) return;
    1125           0 :   err_printf("Rho: time = %6ld ms,\t%3ld round%s\n",
    1126             :              timer_delay(T), c, (c==1?"":"s"));
    1127             : }
    1128             : 
    1129             : static void
    1130    32837089 : one_iter(GEN *x, GEN *P, GEN x1, GEN n, long delta)
    1131             : {
    1132    32837089 :   *x = addis(remii(sqri(*x), n), delta);
    1133    32779440 :   *P = modii(mulii(*P, subii(x1, *x)), n);
    1134    32843283 : }
    1135             : /* Return NULL when we run out of time, or a single t_INT containing a
    1136             :  * nontrivial factor of n, or a vector of t_INTs, each triple of successive
    1137             :  * entries containing a factor, an exponent (equal to one),  and a factor
    1138             :  * class (NULL for unknown or zero for known composite),  matching the
    1139             :  * internal representation used by the ifac_*() routines below. Repeated
    1140             :  * factors may arise; the caller will sort the factors anyway. Result
    1141             :  * is not gerepile-able (contains NULL) */
    1142             : static GEN
    1143        4815 : pollardbrent_i(GEN n, long size, long c0, long retries)
    1144             : {
    1145        4815 :   long tf = lgefint(n), delta, msg_mask, c, k, k1, l;
    1146             :   pari_sp av;
    1147             :   GEN x, x1, y, P, g, g1, res;
    1148             :   pari_timer T;
    1149             : 
    1150        4815 :   if (DEBUGLEVEL >= 4) timer_start(&T);
    1151        4815 :   c = c0 << 5; /* 2^5 iterations per round */
    1152        9630 :   msg_mask = (size >= 448? 0x1fff:
    1153        4815 :                            (size >= 192? (256L<<((size-128)>>6))-1: 0xff));
    1154        4815 :   y = cgeti(tf);
    1155        4815 :   x1= cgeti(tf);
    1156        4815 :   av = avma;
    1157             : 
    1158        4815 : PB_RETRY:
    1159             :  /* trick to make a 'random' choice determined by n.  Don't use x^2+0 or
    1160             :   * x^2-2, ever.  Don't use x^2-3 or x^2-7 with a starting value of 2.
    1161             :   * x^2+4, x^2+9 are affine conjugate to x^2+1, so don't use them either.
    1162             :   *
    1163             :   * (the point being that when we get called again on a composite cofactor
    1164             :   * of something we've already seen, we had better avoid the same delta) */
    1165        4815 :   switch ((size + retries) & 7)
    1166             :   {
    1167        1004 :     case 0:  delta=  1; break;
    1168         672 :     case 1:  delta= -1; break;
    1169         620 :     case 2:  delta=  3; break;
    1170         434 :     case 3:  delta=  5; break;
    1171         545 :     case 4:  delta= -5; break;
    1172         448 :     case 5:  delta=  7; break;
    1173         595 :     case 6:  delta= 11; break;
    1174             :     /* case 7: */
    1175         497 :     default: delta=-11; break;
    1176             :   }
    1177        4815 :   if (DEBUGLEVEL >= 4)
    1178             :   {
    1179           0 :     if (!retries)
    1180           0 :       err_printf("Rho: searching small factor of %ld-bit integer\n", size);
    1181             :     else
    1182           0 :       err_printf("Rho: restarting for remaining rounds...\n");
    1183           0 :     err_printf("Rho: using X^2%+1ld for up to %ld rounds of 32 iterations\n",
    1184             :                delta, c >> 5);
    1185             :   }
    1186        4815 :   x = gen_2; P = gen_1; g1 = NULL; k = 1; l = 1;
    1187        4815 :   affui(2, y);
    1188        4815 :   affui(2, x1);
    1189             :   for (;;) /* terminated under the control of c */
    1190             :   { /* use the polynomial  x^2 + delta */
    1191    15210970 :     one_iter(&x, &P, x1, n, delta);
    1192             : 
    1193    15213382 :     if ((--c & 0x1f)==0)
    1194             :     { /* one round complete */
    1195      471447 :       g = gcdii(n, P); if (!is_pm1(g)) goto fin;
    1196      469745 :       if (c <= 0)
    1197             :       { /* getting bored */
    1198        2155 :         if (DEBUGLEVEL >= 4)
    1199           0 :           err_printf("Rho: time = %6ld ms,\tPollard-Brent giving up.\n",
    1200             :                      timer_delay(&T));
    1201        2155 :         return NULL;
    1202             :       }
    1203      467590 :       P = gen_1;
    1204      467590 :       if (DEBUGLEVEL >= 4) rho_dbg(&T, c0-(c>>5), msg_mask);
    1205      467590 :       affii(x,y); x = y; set_avma(av);
    1206             :     }
    1207             : 
    1208    15207249 :     if (--k) continue; /* normal end of loop body */
    1209             : 
    1210       44281 :     if (c & 0x1f) /* otherwise, we already checked */
    1211             :     {
    1212       28890 :       g = gcdii(n, P); if (!is_pm1(g)) goto fin;
    1213       28869 :       P = gen_1;
    1214             :     }
    1215             : 
    1216             :    /* Fast forward phase, doing l inner iterations without computing gcds.
    1217             :     * Check first whether it would take us beyond the alloted time.
    1218             :     * Fast forward rounds count only half (although they're taking
    1219             :     * more like 2/3 the time of normal rounds).  This to counteract the
    1220             :     * nuisance that all c0 between 4096 and 6144 would act exactly as
    1221             :     * 4096;  with the halving trick only the range 4096..5120 collapses
    1222             :     * (similarly for all other powers of two) */
    1223       44260 :     if ((c -= (l>>1)) <= 0)
    1224             :     { /* got bored */
    1225         997 :       if (DEBUGLEVEL >= 4)
    1226           0 :         err_printf("Rho: time = %6ld ms,\tPollard-Brent giving up.\n",
    1227             :                    timer_delay(&T));
    1228         997 :       return NULL;
    1229             :     }
    1230       43263 :     c &= ~0x1f; /* keep it on multiples of 32 */
    1231             : 
    1232             :     /* Fast forward loop */
    1233       43263 :     affii(x, x1); set_avma(av); x = x1;
    1234       43263 :     k = l; l <<= 1;
    1235             :     /* don't show this for the first several (short) fast forward phases. */
    1236       43263 :     if (DEBUGLEVEL >= 4 && (l>>7) > msg_mask)
    1237           0 :       err_printf("Rho: fast forward phase (%ld rounds of 64)...\n", l>>7);
    1238    17691242 :     for (k1=k; k1; k1--)
    1239             :     {
    1240    17648125 :       one_iter(&x, &P, x1, n, delta);
    1241    17647090 :       if ((k1 & 0x1f) == 0) gerepileall(av, 2, &x, &P);
    1242             :     }
    1243       43117 :     if (DEBUGLEVEL >= 4 && (l>>7) > msg_mask)
    1244           0 :       err_printf("Rho: time = %6ld ms,\t%3ld rounds, back to normal mode\n",
    1245           0 :                  timer_delay(&T), c0-(c>>5));
    1246       43117 :     affii(x,y); P = gerepileuptoint(av, P); x = y;
    1247             :   } /* forever */
    1248             : 
    1249        1663 : fin:
    1250             :   /* An accumulated gcd was > 1 */
    1251        1663 :   if  (!equalii(g,n))
    1252             :   { /* if it isn't n, and looks prime, return it */
    1253        1537 :     if (MR_Jaeschke(g))
    1254             :     {
    1255        1537 :       if (DEBUGLEVEL >= 4)
    1256             :       {
    1257           0 :         rho_dbg(&T, c0-(c>>5), 0);
    1258           0 :         err_printf("\tfound factor = %Ps\n",g);
    1259             :       }
    1260        1537 :       return g;
    1261             :     }
    1262           0 :     set_avma(av); g1 = icopy(g);  /* known composite, keep it safe */
    1263           0 :     av = avma;
    1264             :   }
    1265         126 :   else g1 = n; /* and work modulo g1 for backtracking */
    1266             : 
    1267             :   /* Here g1 is known composite */
    1268         126 :   if (DEBUGLEVEL >= 4 && size > 192)
    1269           0 :     err_printf("Rho: hang on a second, we got something here...\n");
    1270         126 :   x = y;
    1271             :   for(;;)
    1272             :   { /* backtrack until period recovered. Must terminate */
    1273        9450 :     x = addis(remii(sqri(x), g1), delta);
    1274        9450 :     g = gcdii(subii(x1, x), g1); if (!is_pm1(g)) break;
    1275             : 
    1276        9324 :     if (DEBUGLEVEL >= 4 && (--c & 0x1f) == 0) rho_dbg(&T, c0-(c>>5), msg_mask);
    1277             :   }
    1278             : 
    1279         126 :   if (g1 == n || equalii(g,g1))
    1280             :   {
    1281         126 :     if (g1 == n && equalii(g,g1))
    1282             :     { /* out of luck */
    1283           0 :       if (DEBUGLEVEL >= 4)
    1284             :       {
    1285           0 :         rho_dbg(&T, c0-(c>>5), 0);
    1286           0 :         err_printf("\tPollard-Brent failed.\n");
    1287             :       }
    1288           0 :       if (++retries >= 4) pari_err_BUG("");
    1289           0 :       goto PB_RETRY;
    1290             :     }
    1291             :     /* half lucky: we've split n, but g1 equals either g or n */
    1292         126 :     if (DEBUGLEVEL >= 4)
    1293             :     {
    1294           0 :       rho_dbg(&T, c0-(c>>5), 0);
    1295           0 :       err_printf("\tfound %sfactor = %Ps\n", (g1!=n ? "composite " : ""), g);
    1296             :     }
    1297         126 :     res = cgetg(7, t_VEC);
    1298             :     /* g^1: known composite when g1!=n */
    1299         126 :     INIT(res+1, g, gen_1, (g1!=n? gen_0: NULL));
    1300             :     /* cofactor^1: status unknown */
    1301         126 :     INIT(res+4, diviiexact(n,g), gen_1, NULL);
    1302         126 :     return res;
    1303             :   }
    1304             :   /* g < g1 < n : our lucky day -- we've split g1, too */
    1305           0 :   res = cgetg(10, t_VEC);
    1306             :   /* unknown status for all three factors */
    1307           0 :   INIT(res+1, g,                gen_1, NULL);
    1308           0 :   INIT(res+4, diviiexact(g1,g), gen_1, NULL);
    1309           0 :   INIT(res+7, diviiexact(n,g1), gen_1, NULL);
    1310           0 :   if (DEBUGLEVEL >= 4)
    1311             :   {
    1312           0 :     rho_dbg(&T, c0-(c>>5), 0);
    1313           0 :     err_printf("\tfound factors = %Ps, %Ps,\n\tand %Ps\n",
    1314           0 :                gel(res,1), gel(res,4), gel(res,7));
    1315             :   }
    1316           0 :   return res;
    1317             : }
    1318             : /* Tuning parameter:  for input up to 64 bits long, we must not spend more
    1319             :  * than a very short time, for fear of slowing things down on average.
    1320             :  * With the current tuning formula, increase our efforts somewhat at 49 bit
    1321             :  * input (an extra round for each bit at first),  and go up more and more
    1322             :  * rapidly after we pass 80 bits.-- Changed this to adjust for the presence of
    1323             :  * squfof, which will finish input up to 59 bits quickly. */
    1324             : static GEN
    1325        4815 : pollardbrent(GEN n)
    1326             : {
    1327        4815 :   const long tune_pb_min = 14; /* even 15 seems too much. */
    1328        4815 :   long c0, size = expi(n) + 1;
    1329        4815 :   if (size <= 28)
    1330           0 :     c0 = 32;/* amounts very nearly to 'insist'. Now that we have squfof(), we
    1331             :              * don't insist any more when input is 2^29 ... 2^32 */
    1332        4815 :   else if (size <= 42)
    1333        1260 :     c0 = tune_pb_min;
    1334        3555 :   else if (size <= 59) /* match squfof() cutoff point */
    1335        2010 :     c0 = tune_pb_min + ((size - 42)<<1);
    1336        1545 :   else if (size <= 72)
    1337         784 :     c0 = tune_pb_min + size - 24;
    1338         761 :   else if (size <= 301)
    1339             :     /* nonlinear increase in effort, kicking in around 80 bits */
    1340             :     /* 301 gives 48121 + tune_pb_min */
    1341         754 :     c0 = tune_pb_min + size - 60 +
    1342         754 :       ((size-73)>>1)*((size-70)>>3)*((size-56)>>4);
    1343             :   else
    1344           7 :     c0 = 49152; /* ECM is faster when it'd take longer */
    1345        4815 :   return pollardbrent_i(n, size, c0, 0);
    1346             : }
    1347             : GEN
    1348           0 : Z_pollardbrent(GEN n, long rounds, long seed)
    1349             : {
    1350           0 :   pari_sp av = avma;
    1351           0 :   GEN v = pollardbrent_i(n, expi(n)+1, rounds, seed);
    1352           0 :   if (!v) return NULL;
    1353           0 :   if (typ(v) == t_INT) v = mkvec2(v, diviiexact(n,v));
    1354           0 :   else if (lg(v) == 7) v = mkvec2(gel(v,1), gel(v,4));
    1355           0 :   else v = mkvec3(gel(v,1), gel(v,4), gel(v,7));
    1356           0 :   return gerepilecopy(av, v);
    1357             : }
    1358             : 
    1359             : /***********************************************************************/
    1360             : /**              FACTORIZATION (Shanks' SQUFOF) --GN2000Sep30-Oct01   **/
    1361             : /**  squfof() returns a nontrivial factor of n, assuming n is odd,    **/
    1362             : /**  composite, not a pure square, and has no small prime divisor,    **/
    1363             : /**  or NULL if it fails to find one.  It works on two discriminants  **/
    1364             : /**  simultaneously  (n and 5n for n=1(4), 3n and 4n for n=3(4)).     **/
    1365             : /**  Present implementation is limited to input <2^59, and works most **/
    1366             : /**  of the time in signed arithmetic on integers <2^31 in absolute   **/
    1367             : /**  size. (Cf. Algo 8.7.2 in ACiCNT)                                 **/
    1368             : /***********************************************************************/
    1369             : 
    1370             : /* The following is invoked to walk back along the ambiguous cycle* until we
    1371             :  * hit an ambiguous form and thus the desired factor, which it returns.  If it
    1372             :  * fails for any reason, it returns 0.  It doesn't interfere with timing and
    1373             :  * diagnostics, which it leaves to squfof().
    1374             :  *
    1375             :  * Before we invoke this, we've found a form (A, B, -C) with A = a^2, where a
    1376             :  * isn't blacklisted and where gcd(a, B) = 1.  According to ACiCANT, we should
    1377             :  * now proceed reducing the form (a, -B, -aC), but it is easy to show that the
    1378             :  * first reduction step always sends this to (-aC, B, a), and the next one,
    1379             :  * with q computed as usual from B and a (occupying the c position), gives a
    1380             :  * reduced form, whose third member is easiest to recover by going back to D.
    1381             :  * From this point onwards, we're once again working with single-word numbers.
    1382             :  * No need to track signs, just work with the abs values of the coefficients. */
    1383             : static long
    1384        2843 : squfof_ambig(long a, long B, long dd, GEN D)
    1385             : {
    1386             :   long b, c, q, qa, qc, qcb, a0, b0, b1, c0;
    1387        2843 :   long cnt = 0; /* count reduction steps on the cycle */
    1388             : 
    1389        2843 :   q = (dd + (B>>1)) / a;
    1390        2843 :   qa = q * a;
    1391        2843 :   b = (qa - B) + qa; /* avoid overflow */
    1392             :   {
    1393        2843 :     pari_sp av = avma;
    1394        2843 :     c = itos(divis(shifti(subii(D, sqrs(b)), -2), a));
    1395        2843 :     set_avma(av);
    1396             :   }
    1397             : #ifdef DEBUG_SQUFOF
    1398             :   err_printf("SQUFOF: ambigous cycle of discriminant %Ps\n", D);
    1399             :   err_printf("SQUFOF: Form on ambigous cycle (%ld, %ld, %ld)\n", a, b, c);
    1400             : #endif
    1401             : 
    1402        2843 :   a0 = a; b0 = b1 = b;        /* end of loop detection and safeguard */
    1403             : 
    1404             :   for (;;) /* reduced cycles are finite */
    1405             :   { /* reduction step */
    1406     5165938 :     c0 = c;
    1407     5165938 :     if (c0 > dd)
    1408     1443038 :       q = 1;
    1409             :     else
    1410     3722900 :       q = (dd + (b>>1)) / c0;
    1411     5165938 :     if (q == 1)
    1412             :     {
    1413     2148918 :       qcb = c0 - b; b = c0 + qcb; c = a - qcb;
    1414             :     }
    1415             :     else
    1416             :     {
    1417     3017020 :       qc = q*c0; qcb = qc - b; b = qc + qcb; c = a - q*qcb;
    1418             :     }
    1419     5165938 :     a = c0;
    1420             : 
    1421     5165938 :     cnt++; if (b == b1) break;
    1422             : 
    1423             :     /* safeguard against infinite loop: recognize when we've walked the entire
    1424             :      * cycle in vain. (I don't think this can actually happen -- exercise.) */
    1425     5163095 :     if (b == b0 && a == a0) return 0;
    1426             : 
    1427     5163095 :     b1 = b;
    1428             :   }
    1429        2843 :   q = a&1 ? a : a>>1;
    1430        2843 :   if (DEBUGLEVEL >= 4)
    1431             :   {
    1432           0 :     if (q > 1)
    1433           0 :       err_printf("SQUFOF: found factor %ld from ambiguous form\n"
    1434             :                  "\tafter %ld steps on the ambiguous cycle\n",
    1435           0 :                  q / ugcd(q,15), cnt);
    1436             :     else
    1437           0 :       err_printf("SQUFOF: ...found nothing on the ambiguous cycle\n"
    1438             :                  "\tafter %ld steps there\n", cnt);
    1439           0 :     if (DEBUGLEVEL >= 6) err_printf("SQUFOF: squfof_ambig returned %ld\n", q);
    1440             :   }
    1441        2843 :   return q;
    1442             : }
    1443             : 
    1444             : #define SQUFOF_BLACKLIST_SZ 64
    1445             : 
    1446             : /* assume 2,3,5 do not divide n */
    1447             : static GEN
    1448        3152 : squfof(GEN n)
    1449             : {
    1450             :   ulong d1, d2;
    1451        3152 :   long tf = lgefint(n), nm4, cnt = 0;
    1452             :   long a1, b1, c1, dd1, L1, a2, b2, c2, dd2, L2, a, q, c, qc, qcb;
    1453             :   GEN D1, D2;
    1454        3152 :   pari_sp av = avma;
    1455             :   long blacklist1[SQUFOF_BLACKLIST_SZ], blacklist2[SQUFOF_BLACKLIST_SZ];
    1456        3152 :   long blp1 = 0, blp2 = 0;
    1457        3152 :   int act1 = 1, act2 = 1;
    1458             : 
    1459             : #ifdef LONG_IS_64BIT
    1460        2736 :   if (tf > 3 || (tf == 3 && uel(n,2)             >= (1UL << (BITS_IN_LONG-5))))
    1461             : #else  /* 32 bits */
    1462         416 :   if (tf > 4 || (tf == 4 && (ulong)(*int_MSW(n)) >= (1UL << (BITS_IN_LONG-5))))
    1463             : #endif
    1464         939 :     return NULL; /* n too large */
    1465             : 
    1466             :   /* now we have 5 < n < 2^59 */
    1467        2213 :   nm4 = mod4(n);
    1468        2213 :   if (nm4 == 1)
    1469             :   { /* n = 1 (mod4):  run one iteration on D1 = n, another on D2 = 5n */
    1470        1103 :     D1 = n;
    1471        1103 :     D2 = mului(5,n); d2 = itou(sqrti(D2)); dd2 = (long)((d2>>1) + (d2&1));
    1472        1103 :     b2 = (long)((d2-1) | 1);        /* b1, b2 will always stay odd */
    1473             :   }
    1474             :   else
    1475             :   { /* n = 3 (mod4):  run one iteration on D1 = 3n, another on D2 = 4n */
    1476        1110 :     D1 = mului(3,n);
    1477        1110 :     D2 = shifti(n,2); dd2 = itou(sqrti(n)); d2 =  dd2 << 1;
    1478        1110 :     b2 = (long)(d2 & (~1UL)); /* largest even below d2, will stay even */
    1479             :   }
    1480        2213 :   d1 = itou(sqrti(D1));
    1481        2213 :   b1 = (long)((d1-1) | 1); /* largest odd number not exceeding d1 */
    1482        2213 :   c1 = itos(shifti(subii(D1, sqru((ulong)b1)), -2));
    1483        2213 :   if (!c1) pari_err_BUG("squfof [caller of] (n or 3n is a square)");
    1484        2213 :   c2 = itos(shifti(subii(D2, sqru((ulong)b2)), -2));
    1485        2213 :   if (!c2) pari_err_BUG("squfof [caller of] (5n is a square)");
    1486        2213 :   L1 = (long)usqrt(d1);
    1487        2213 :   L2 = (long)usqrt(d2);
    1488             :   /* dd1 used to compute floor((d1+b1)/2) as dd1+floor(b1/2), without
    1489             :    * overflowing the 31bit signed integer size limit. Same for dd2. */
    1490        2213 :   dd1 = (long) ((d1>>1) + (d1&1));
    1491        2213 :   a1 = a2 = 1;
    1492             : 
    1493             :   /* The two (identity) forms (a1,b1,-c1) and (a2,b2,-c2) are now set up.
    1494             :    *
    1495             :    * a1 and c1 represent the absolute values of the a,c coefficients; we keep
    1496             :    * track of the sign separately, via the iteration counter cnt: when cnt is
    1497             :    * even, c is understood to be negative, else c is positive and a < 0.
    1498             :    *
    1499             :    * L1, L2 are the limits for blacklisting small leading coefficients
    1500             :    * on the principal cycle, to guarantee that when we find a square form,
    1501             :    * its square root will belong to an ambiguous cycle  (i.e. won't be an
    1502             :    * earlier form on the principal cycle).
    1503             :    *
    1504             :    * When n = 3(mod 4), D2 = 12(mod 16), and b^2 is always 0 or 4 mod 16.
    1505             :    * It follows that 4*a*c must be 4 or 8 mod 16, respectively, so at most
    1506             :    * one of a,c can be divisible by 2 at most to the first power.  This fact
    1507             :    * is used a couple of times below.
    1508             :    *
    1509             :    * The flags act1, act2 remain true while the respective cycle is still
    1510             :    * active;  we drop them to false when we return to the identity form with-
    1511             :    * out having found a square form  (or when the blacklist overflows, which
    1512             :    * shouldn't happen). */
    1513        2213 :   if (DEBUGLEVEL >= 4)
    1514           0 :     err_printf("SQUFOF: entering main loop with forms\n"
    1515             :                "\t(1, %ld, %ld) and (1, %ld, %ld)\n\tof discriminants\n"
    1516             :                "\t%Ps and %Ps, respectively\n", b1, -c1, b2, -c2, D1, D2);
    1517             : 
    1518             :   /* MAIN LOOP: walk around the principal cycle looking for a square form.
    1519             :    * Blacklist small leading coefficients.
    1520             :    *
    1521             :    * The reduction operator can be computed entirely in 32-bit arithmetic:
    1522             :    * Let q = floor(floor((d1+b1)/2)/c1)  (when c1>dd1, q=1, which happens
    1523             :    * often enough to special-case it).  Then the new b1 = (q*c1-b1) + q*c1,
    1524             :    * which does not overflow, and the new c1 = a1 - q*(q*c1-b1), which is
    1525             :    * bounded by d1 in abs size since both the old and the new a1 are positive
    1526             :    * and bounded by d1. */
    1527     7687283 :   while (act1 || act2)
    1528             :   {
    1529     7687276 :     if (act1)
    1530             :     { /* send first form through reduction operator if active */
    1531     7687192 :       c = c1;
    1532     7687192 :       q = (c > dd1)? 1: (dd1 + (b1>>1)) / c;
    1533     7687192 :       if (q == 1)
    1534     3182735 :       { qcb = c - b1; b1 = c + qcb; c1 = a1 - qcb; }
    1535             :       else
    1536     4504457 :       { qc = q*c; qcb = qc - b1; b1 = qc + qcb; c1 = a1 - q*qcb; }
    1537     7687192 :       a1 = c;
    1538             : 
    1539     7687192 :       if (a1 <= L1)
    1540             :       { /* blacklist this */
    1541        2079 :         if (blp1 >= SQUFOF_BLACKLIST_SZ) /* overflows: shouldn't happen */
    1542           0 :           act1 = 0; /* silently */
    1543             :         else
    1544             :         {
    1545        2079 :           if (DEBUGLEVEL >= 6)
    1546           0 :             err_printf("SQUFOF: blacklisting a = %ld on first cycle\n", a1);
    1547        2079 :           blacklist1[blp1++] = a1;
    1548             :         }
    1549             :       }
    1550             :     }
    1551     7687276 :     if (act2)
    1552             :     { /* send second form through reduction operator if active */
    1553     7686226 :       c = c2;
    1554     7686226 :       q = (c > dd2)? 1: (dd2 + (b2>>1)) / c;
    1555     7686226 :       if (q == 1)
    1556     3192113 :       { qcb = c - b2; b2 = c + qcb; c2 = a2 - qcb; }
    1557             :       else
    1558     4494113 :       { qc = q*c; qcb = qc - b2; b2 = qc + qcb; c2 = a2 - q*qcb; }
    1559     7686226 :       a2 = c;
    1560             : 
    1561     7686226 :       if (a2 <= L2)
    1562             :       { /* blacklist this */
    1563        1730 :         if (blp2 >= SQUFOF_BLACKLIST_SZ) /* overflows: shouldn't happen */
    1564           0 :           act2 = 0; /* silently */
    1565             :         else
    1566             :         {
    1567        1730 :           if (DEBUGLEVEL >= 6)
    1568           0 :             err_printf("SQUFOF: blacklisting a = %ld on second cycle\n", a2);
    1569        1730 :           blacklist2[blp2++] = a2;
    1570             :         }
    1571             :       }
    1572             :     }
    1573             : 
    1574             :     /* bump counter, loop if this is an odd iteration (i.e. if the real
    1575             :      * leading coefficients are negative) */
    1576     7687276 :     if (++cnt & 1) continue;
    1577             : 
    1578             :     /* second half of main loop entered only when the leading coefficients
    1579             :      * are positive (i.e., during even-numbered iterations) */
    1580             : 
    1581             :     /* examine first form if active */
    1582     3843638 :     if (act1 && a1 == 1) /* back to identity */
    1583             :     { /* drop this discriminant */
    1584           7 :       act1 = 0;
    1585           7 :       if (DEBUGLEVEL >= 4)
    1586           0 :         err_printf("SQUFOF: first cycle exhausted after %ld iterations,\n"
    1587             :                    "\tdropping it\n", cnt);
    1588             :     }
    1589     3843638 :     if (act1)
    1590             :     {
    1591     3843589 :       if (uissquareall((ulong)a1, (ulong*)&a))
    1592             :       { /* square form */
    1593        2148 :         if (DEBUGLEVEL >= 4)
    1594           0 :           err_printf("SQUFOF: square form (%ld^2, %ld, %ld) on first cycle\n"
    1595             :                      "\tafter %ld iterations\n", a, b1, -c1, cnt);
    1596        2148 :         if (a <= L1)
    1597             :         { /* blacklisted? */
    1598             :           long j;
    1599        4367 :           for (j = 0; j < blp1; j++)
    1600        3157 :             if (a == blacklist1[j]) { a = 0; break; }
    1601             :         }
    1602        2148 :         if (a > 0)
    1603             :         { /* not blacklisted */
    1604        1210 :           q = ugcd(a, b1); /* imprimitive form? */
    1605        1210 :           if (q > 1)
    1606             :           { /* q^2 divides D1 hence n [ assuming n % 3 != 0 ] */
    1607           0 :             set_avma(av);
    1608           0 :             if (DEBUGLEVEL >= 4) err_printf("SQUFOF: found factor %ld^2\n", q);
    1609           0 :             return mkvec3(utoipos(q), gen_2, NULL);/* exponent 2, unknown status */
    1610             :           }
    1611             :           /* chase the inverse root form back along the ambiguous cycle */
    1612        1210 :           q = squfof_ambig(a, b1, dd1, D1);
    1613        1210 :           if (nm4 == 3 && q % 3 == 0) q /= 3;
    1614        1210 :           if (q > 1) return gc_utoipos(av, q); /* SUCCESS! */
    1615             :         }
    1616         938 :         else if (DEBUGLEVEL >= 4) /* blacklisted */
    1617           0 :           err_printf("SQUFOF: ...but the root form seems to be on the "
    1618             :                      "principal cycle\n");
    1619             :       }
    1620             :     }
    1621             : 
    1622             :     /* examine second form if active */
    1623     3842673 :     if (act2 && a2 == 1) /* back to identity form */
    1624             :     { /* drop this discriminant */
    1625          14 :       act2 = 0;
    1626          14 :       if (DEBUGLEVEL >= 4)
    1627           0 :         err_printf("SQUFOF: second cycle exhausted after %ld iterations,\n"
    1628             :                    "\tdropping it\n", cnt);
    1629             :     }
    1630     3842673 :     if (act2)
    1631             :     {
    1632     3842141 :       if (uissquareall((ulong)a2, (ulong*)&a))
    1633             :       { /* square form */
    1634        2109 :         if (DEBUGLEVEL >= 4)
    1635           0 :           err_printf("SQUFOF: square form (%ld^2, %ld, %ld) on second cycle\n"
    1636             :                      "\tafter %ld iterations\n", a, b2, -c2, cnt);
    1637        2109 :         if (a <= L2)
    1638             :         { /* blacklisted? */
    1639             :           long j;
    1640        3727 :           for (j = 0; j < blp2; j++)
    1641        2094 :             if (a == blacklist2[j]) { a = 0; break; }
    1642             :         }
    1643        2109 :         if (a > 0)
    1644             :         { /* not blacklisted */
    1645        1633 :           q = ugcd(a, b2); /* imprimitive form? */
    1646             :           /* NB if b2 is even, a is odd, so the gcd is always odd */
    1647        1633 :           if (q > 1)
    1648             :           { /* q^2 divides D2 hence n [ assuming n % 5 != 0 ] */
    1649           0 :             set_avma(av);
    1650           0 :             if (DEBUGLEVEL >= 4) err_printf("SQUFOF: found factor %ld^2\n", q);
    1651           0 :             return mkvec3(utoipos(q), gen_2, NULL);/* exponent 2, unknown status */
    1652             :           }
    1653             :           /* chase the inverse root form along the ambiguous cycle */
    1654        1633 :           q = squfof_ambig(a, b2, dd2, D2);
    1655        1633 :           if (nm4 == 1 && q % 5 == 0) q /= 5;
    1656        1633 :           if (q > 1) return gc_utoipos(av, q); /* SUCCESS! */
    1657             :         }
    1658         476 :         else if (DEBUGLEVEL >= 4)        /* blacklisted */
    1659           0 :           err_printf("SQUFOF: ...but the root form seems to be on the "
    1660             :                      "principal cycle\n");
    1661             :       }
    1662             :     }
    1663             :   } /* end main loop */
    1664             : 
    1665             :   /* both discriminants turned out to be useless. */
    1666           7 :   if (DEBUGLEVEL>=4) err_printf("SQUFOF: giving up\n");
    1667           7 :   return gc_NULL(av);
    1668             : }
    1669             : 
    1670             : /***********************************************************************/
    1671             : /*                    DETECTING ODD POWERS  --GN1998Jun28              */
    1672             : /*   Factoring engines like MPQS which ultimately rely on computing    */
    1673             : /*   gcd(N, x^2-y^2) to find a nontrivial factor of N can't split      */
    1674             : /*   N = p^k for an odd prime p, since (Z/p^k)^* is then cyclic. Here  */
    1675             : /*   is an analogue of Z_issquareall() for 3rd, 5th and 7th powers.    */
    1676             : /*   The general case is handled by is_kth_power                       */
    1677             : /***********************************************************************/
    1678             : 
    1679             : /* Multistage sieve. First stages work mod 211, 209, 61, 203 in this order
    1680             :  * (first reduce mod the product of these and then take the remainder apart).
    1681             :  * Second stages use 117, 31, 43, 71. Moduli which are no longer interesting
    1682             :  * are skipped. Everything is encoded in a table of 106 24-bit masks. We only
    1683             :  * need the first half of the residues.  Three bits per modulus indicate which
    1684             :  * residues are 7th (bit 2), 5th (bit 1) or 3rd (bit 0) powers; the eight
    1685             :  * moduli above are assigned right-to-left. The table was generated using: */
    1686             : 
    1687             : #if 0
    1688             : L = [71, 43, 31, [O(3^2),O(13)], [O(7),O(29)], 61, [O(11),O(19)], 211];
    1689             : ispow(x, N, k)=
    1690             : {
    1691             :   if (type(N) == "t_INT", return (ispower(Mod(x,N), k)));
    1692             :   for (i = 1, #N, if (!ispower(x + N[i], k), return (0))); 1
    1693             : }
    1694             : check(r) =
    1695             : {
    1696             :   print1("  0");
    1697             :   for (i=1,#L,
    1698             :     N = 0;
    1699             :     if (ispow(r, L[i], 3), N += 1);
    1700             :     if (ispow(r, L[i], 5), N += 2);
    1701             :     if (ispow(r, L[i], 7), N += 4);
    1702             :     print1(N);
    1703             :   ); print("ul,  /* ", r, " */")
    1704             : }
    1705             : for (r = 0, 105, check(r))
    1706             : #endif
    1707             : static ulong powersmod[106] = {
    1708             :   077777777ul,  /* 0 */
    1709             :   077777777ul,  /* 1 */
    1710             :   013562440ul,  /* 2 */
    1711             :   012402540ul,  /* 3 */
    1712             :   013562440ul,  /* 4 */
    1713             :   052662441ul,  /* 5 */
    1714             :   016603440ul,  /* 6 */
    1715             :   016463450ul,  /* 7 */
    1716             :   013573551ul,  /* 8 */
    1717             :   012462540ul,  /* 9 */
    1718             :   012462464ul,  /* 10 */
    1719             :   013462771ul,  /* 11 */
    1720             :   012406473ul,  /* 12 */
    1721             :   012463641ul,  /* 13 */
    1722             :   052463646ul,  /* 14 */
    1723             :   012503446ul,  /* 15 */
    1724             :   013562440ul,  /* 16 */
    1725             :   052466440ul,  /* 17 */
    1726             :   012472451ul,  /* 18 */
    1727             :   012462454ul,  /* 19 */
    1728             :   032463550ul,  /* 20 */
    1729             :   013403664ul,  /* 21 */
    1730             :   013463460ul,  /* 22 */
    1731             :   032562565ul,  /* 23 */
    1732             :   012402540ul,  /* 24 */
    1733             :   052662441ul,  /* 25 */
    1734             :   032672452ul,  /* 26 */
    1735             :   013573551ul,  /* 27 */
    1736             :   012467541ul,  /* 28 */
    1737             :   012567640ul,  /* 29 */
    1738             :   032706450ul,  /* 30 */
    1739             :   012762452ul,  /* 31 */
    1740             :   033762662ul,  /* 32 */
    1741             :   012502562ul,  /* 33 */
    1742             :   032463562ul,  /* 34 */
    1743             :   013563440ul,  /* 35 */
    1744             :   016663440ul,  /* 36 */
    1745             :   036662550ul,  /* 37 */
    1746             :   012462552ul,  /* 38 */
    1747             :   033502450ul,  /* 39 */
    1748             :   012462643ul,  /* 40 */
    1749             :   033467540ul,  /* 41 */
    1750             :   017403441ul,  /* 42 */
    1751             :   017463462ul,  /* 43 */
    1752             :   017472460ul,  /* 44 */
    1753             :   033462470ul,  /* 45 */
    1754             :   052566450ul,  /* 46 */
    1755             :   013562640ul,  /* 47 */
    1756             :   032403640ul,  /* 48 */
    1757             :   016463450ul,  /* 49 */
    1758             :   016463752ul,  /* 50 */
    1759             :   033402440ul,  /* 51 */
    1760             :   012462540ul,  /* 52 */
    1761             :   012472540ul,  /* 53 */
    1762             :   053562462ul,  /* 54 */
    1763             :   012463465ul,  /* 55 */
    1764             :   012663470ul,  /* 56 */
    1765             :   052607450ul,  /* 57 */
    1766             :   012566553ul,  /* 58 */
    1767             :   013466440ul,  /* 59 */
    1768             :   012502741ul,  /* 60 */
    1769             :   012762744ul,  /* 61 */
    1770             :   012763740ul,  /* 62 */
    1771             :   012763443ul,  /* 63 */
    1772             :   013573551ul,  /* 64 */
    1773             :   013462471ul,  /* 65 */
    1774             :   052502460ul,  /* 66 */
    1775             :   012662463ul,  /* 67 */
    1776             :   012662451ul,  /* 68 */
    1777             :   012403550ul,  /* 69 */
    1778             :   073567540ul,  /* 70 */
    1779             :   072463445ul,  /* 71 */
    1780             :   072462740ul,  /* 72 */
    1781             :   012472442ul,  /* 73 */
    1782             :   012462644ul,  /* 74 */
    1783             :   013406650ul,  /* 75 */
    1784             :   052463471ul,  /* 76 */
    1785             :   012563474ul,  /* 77 */
    1786             :   013503460ul,  /* 78 */
    1787             :   016462441ul,  /* 79 */
    1788             :   016462440ul,  /* 80 */
    1789             :   012462540ul,  /* 81 */
    1790             :   013462641ul,  /* 82 */
    1791             :   012463454ul,  /* 83 */
    1792             :   013403550ul,  /* 84 */
    1793             :   057563540ul,  /* 85 */
    1794             :   017466441ul,  /* 86 */
    1795             :   017606471ul,  /* 87 */
    1796             :   053666573ul,  /* 88 */
    1797             :   012562561ul,  /* 89 */
    1798             :   013473641ul,  /* 90 */
    1799             :   032573440ul,  /* 91 */
    1800             :   016763440ul,  /* 92 */
    1801             :   016702640ul,  /* 93 */
    1802             :   033762552ul,  /* 94 */
    1803             :   012562550ul,  /* 95 */
    1804             :   052402451ul,  /* 96 */
    1805             :   033563441ul,  /* 97 */
    1806             :   012663561ul,  /* 98 */
    1807             :   012677560ul,  /* 99 */
    1808             :   012462464ul,  /* 100 */
    1809             :   032562642ul,  /* 101 */
    1810             :   013402551ul,  /* 102 */
    1811             :   032462450ul,  /* 103 */
    1812             :   012467445ul,  /* 104 */
    1813             :   032403440ul,  /* 105 */
    1814             : };
    1815             : 
    1816             : static int
    1817     4060828 : check_res(ulong x, ulong N, int shift, ulong *mask)
    1818             : {
    1819     4060828 :   long r = x%N; if ((ulong)r> (N>>1)) r = N - r;
    1820     4060828 :   *mask &= (powersmod[r] >> shift);
    1821     4060828 :   return *mask;
    1822             : }
    1823             : 
    1824             : /* is x mod 211*209*61*203*117*31*43*71 a 3rd, 5th or 7th power ? */
    1825             : int
    1826     2510935 : uis_357_powermod(ulong x, ulong *mask)
    1827             : {
    1828     2510935 :   if (             !check_res(x, 211UL, 0, mask)) return 0;
    1829     1008036 :   if (*mask & 3 && !check_res(x, 209UL, 3, mask)) return 0;
    1830      409184 :   if (*mask & 3 && !check_res(x,  61UL, 6, mask)) return 0;
    1831      225314 :   if (*mask & 5 && !check_res(x, 203UL, 9, mask)) return 0;
    1832       58096 :   if (*mask & 1 && !check_res(x, 117UL,12, mask)) return 0;
    1833       32852 :   if (*mask & 3 && !check_res(x,  31UL,15, mask)) return 0;
    1834       24633 :   if (*mask & 5 && !check_res(x,  43UL,18, mask)) return 0;
    1835        7423 :   if (*mask & 6 && !check_res(x,  71UL,21, mask)) return 0;
    1836        3763 :   return 1;
    1837             : }
    1838             : /* asume x > 0 and pt != NULL */
    1839             : int
    1840     2456114 : uis_357_power(ulong x, ulong *pt, ulong *mask)
    1841             : {
    1842             :   double logx;
    1843     2456114 :   if (!odd(x))
    1844             :   {
    1845        9074 :     long v = vals(x);
    1846        9074 :     if (v % 7) *mask &= ~4;
    1847        9074 :     if (v % 5) *mask &= ~2;
    1848        9074 :     if (v % 3) *mask &= ~1;
    1849        9074 :     if (!*mask) return 0;
    1850             :   }
    1851     2448528 :   if (!uis_357_powermod(x, mask)) return 0;
    1852        2970 :   logx = log((double)x);
    1853        3864 :   while (*mask)
    1854             :   {
    1855             :     long e, b;
    1856             :     ulong y, ye;
    1857        2970 :     if (*mask & 1)      { b = 1; e = 3; }
    1858         866 :     else if (*mask & 2) { b = 2; e = 5; }
    1859         355 :     else                { b = 4; e = 7; }
    1860        2970 :     y = (ulong)(exp(logx / e) + 0.5);
    1861        2970 :     ye = upowuu(y,e);
    1862        2970 :     if (ye == x) { *pt = y; return e; }
    1863             : #ifdef LONG_IS_64BIT
    1864         768 :     if (ye > x) y--; else y++;
    1865         768 :     ye = upowuu(y,e);
    1866         768 :     if (ye == x) { *pt = y; return e; }
    1867             : #endif
    1868         894 :     *mask &= ~b; /* turn the bit off */
    1869             :   }
    1870         894 :   return 0;
    1871             : }
    1872             : 
    1873             : #ifndef LONG_IS_64BIT
    1874             : /* as above, split in two functions */
    1875             : /* is x mod 211*209*61*203 a 3rd, 5th or 7th power ? */
    1876             : static int
    1877       14110 : uis_357_powermod_32bit_1(ulong x, ulong *mask)
    1878             : {
    1879       14110 :   if (             !check_res(x, 211UL, 0, mask)) return 0;
    1880        7839 :   if (*mask & 3 && !check_res(x, 209UL, 3, mask)) return 0;
    1881        4033 :   if (*mask & 3 && !check_res(x,  61UL, 6, mask)) return 0;
    1882        2861 :   if (*mask & 5 && !check_res(x, 203UL, 9, mask)) return 0;
    1883         813 :   return 1;
    1884             : }
    1885             : /* is x mod 117*31*43*71 a 3rd, 5th or 7th power ? */
    1886             : static int
    1887         813 : uis_357_powermod_32bit_2(ulong x, ulong *mask)
    1888             : {
    1889         813 :   if (*mask & 1 && !check_res(x, 117UL,12, mask)) return 0;
    1890         646 :   if (*mask & 3 && !check_res(x,  31UL,15, mask)) return 0;
    1891         528 :   if (*mask & 5 && !check_res(x,  43UL,18, mask)) return 0;
    1892         263 :   if (*mask & 6 && !check_res(x,  71UL,21, mask)) return 0;
    1893         174 :   return 1;
    1894             : }
    1895             : #endif
    1896             : 
    1897             : /* Returns 3, 5, or 7 if x is a cube (but not a 5th or 7th power),  a 5th
    1898             :  * power (but not a 7th),  or a 7th power, and in this case creates the
    1899             :  * base on the stack and assigns its address to *pt.  Otherwise returns 0.
    1900             :  * x must be of type t_INT and positive;  this is not checked.  The *mask
    1901             :  * argument tells us which things to check -- bit 0: 3rd, bit 1: 5th,
    1902             :  * bit 2: 7th pwr;  set a bit to have the corresponding power examined --
    1903             :  * and is updated appropriately for a possible follow-up call */
    1904             : int
    1905     2793938 : is_357_power(GEN x, GEN *pt, ulong *mask)
    1906             : {
    1907     2793938 :   long lx = lgefint(x);
    1908             :   ulong r;
    1909             :   pari_sp av;
    1910             :   GEN y;
    1911             : 
    1912     2793938 :   if (!*mask) return 0; /* useful when running in a loop */
    1913     2424009 :   if (DEBUGLEVEL>4)
    1914           0 :     err_printf("OddPwrs: examining %ld-bit integer\n", expi(x)+1);
    1915     2424009 :   if (lgefint(x) == 3) {
    1916             :     ulong t;
    1917     2347492 :     long e = uis_357_power(x[2], &t, mask);
    1918     2347492 :     if (e)
    1919             :     {
    1920        2050 :       if (pt) *pt = utoi(t);
    1921        2050 :       return e;
    1922             :     }
    1923     2345442 :     return 0;
    1924             :   }
    1925             : #ifdef LONG_IS_64BIT
    1926       62407 :   r = (lx == 3)? uel(x,2): umodiu(x, 6046846918939827UL);
    1927       62407 :   if (!uis_357_powermod(r, mask)) return 0;
    1928             : #else
    1929       14110 :   r = (lx == 3)? uel(x,2): umodiu(x, 211*209*61*203);
    1930       14110 :   if (!uis_357_powermod_32bit_1(r, mask)) return 0;
    1931         813 :   r = (lx == 3)? uel(x,2): umodiu(x, 117*31*43*71);
    1932         813 :   if (!uis_357_powermod_32bit_2(r, mask)) return 0;
    1933             : #endif
    1934         967 :   av = avma;
    1935        1588 :   while (*mask)
    1936             :   {
    1937             :     long e, b;
    1938             :     /* priority to higher powers: if we have a 21st, it is easier to rediscover
    1939             :      * that its 7th root is a cube than that its cube root is a 7th power */
    1940        1289 :          if (*mask & 4) { b = 4; e = 7; }
    1941         949 :     else if (*mask & 2) { b = 2; e = 5; }
    1942         359 :     else                { b = 1; e = 3; }
    1943        1289 :     y = mpround( sqrtnr(itor(x, nbits2prec(64 + bit_accuracy(lx) / e)), e) );
    1944        1289 :     if (equalii(powiu(y,e), x))
    1945             :     {
    1946         668 :       if (!pt) return gc_int(av,e);
    1947         668 :       set_avma((pari_sp)y); *pt = gerepileuptoint(av, y);
    1948         668 :       return e;
    1949             :     }
    1950         621 :     *mask &= ~b; /* turn the bit off */
    1951         621 :     set_avma(av);
    1952             :   }
    1953         299 :   return 0;
    1954             : }
    1955             : 
    1956             : /* Is x a n-th power ?
    1957             :  * if d = NULL, n not necessarily prime, otherwise, n prime and d the
    1958             :  * corresponding diffptr to go on looping over primes.
    1959             :  * If pt != NULL, it receives the n-th root */
    1960             : ulong
    1961      487831 : is_kth_power(GEN x, ulong n, GEN *pt)
    1962             : {
    1963             :   forprime_t T;
    1964             :   long j;
    1965             :   ulong q, residue;
    1966             :   GEN y;
    1967      487831 :   pari_sp av = avma;
    1968             : 
    1969      487831 :   (void)u_forprime_arith_init(&T, odd(n)? 2*n+1: n+1, ULONG_MAX, 1,n);
    1970             :   /* we'll start at q, smallest prime >= n */
    1971             : 
    1972             :   /* Modular checks, use small primes q congruent 1 mod n */
    1973             :   /* A non n-th power nevertheless passes the test with proba n^(-#checks),
    1974             :    * We'd like this < 1e-6 but let j = floor(log(1e-6) / log(n)) which
    1975             :    * ensures much less. */
    1976      487858 :   if (n < 16)
    1977      118429 :     j = 5;
    1978      369429 :   else if (n < 32)
    1979      155591 :     j = 4;
    1980      213838 :   else if (n < 101)
    1981      190237 :     j = 3;
    1982       23601 :   else if (n < 1001)
    1983        6972 :     j = 2;
    1984       16629 :   else if (n < 17886697) /* smallest such that smallest suitable q is > 2^32 */
    1985       16303 :     j = 1;
    1986             :   else
    1987         326 :     j = 0;
    1988      531679 :   for (; j > 0; j--)
    1989             :   {
    1990      529072 :     if (!(q = u_forprime_next(&T))) break;
    1991             :     /* q a prime = 1 mod n */
    1992      529063 :     residue = umodiu(x, q);
    1993      529069 :     if (residue == 0)
    1994             :     {
    1995         490 :       if (Z_lval(x,q) % n) return gc_ulong(av,0);
    1996          49 :       continue;
    1997             :     }
    1998             :     /* n-th power mod q ? */
    1999      528579 :     if (Fl_powu(residue, (q-1)/n, q) != 1) return gc_ulong(av,0);
    2000             :   }
    2001        2607 :   set_avma(av);
    2002             : 
    2003        2604 :   if (DEBUGLEVEL>4) err_printf("\nOddPwrs: [%lu] passed modular checks\n",n);
    2004             :   /* go to the horse's mouth... */
    2005        2604 :   y = roundr( sqrtnr(itor(x, nbits2prec(expi(x)/n + 16)), n) );
    2006        2604 :   if (!equalii(powiu(y, n), x)) {
    2007        1662 :     if (DEBUGLEVEL>4) err_printf("\tBut it wasn't a pure power.\n");
    2008        1662 :     return gc_ulong(av,0);
    2009             :   }
    2010         942 :   if (!pt) set_avma(av); else { set_avma((pari_sp)y); *pt = gerepileuptoint(av,y); }
    2011         942 :   return 1;
    2012             : }
    2013             : 
    2014             : /* is x a p^i-th power, p >= 11 prime ? Similar to is_357_power(), but instead
    2015             :  * of the mask, we keep the current test exponent around. Cut off when
    2016             :  * log_2 x^(1/k) < cutoffbits since we would have found it by trial division.
    2017             :  * Everything needed here (primitive roots etc.) is computed from scratch on
    2018             :  * the fly; compared to the size of numbers under consideration, these
    2019             :  * word-sized computations take negligible time.
    2020             :  * Any cutoffbits > 0 is safe, but direct root extraction attempts are faster
    2021             :  * when trial division has been used to discover very small bases. We become
    2022             :  * competitive at cutoffbits ~ 10 */
    2023             : int
    2024       69580 : is_pth_power(GEN x, GEN *pt, forprime_t *T, ulong cutoffbits)
    2025             : {
    2026       69580 :   long cnt=0, size = expi(x) /* not +1 */;
    2027             :   ulong p;
    2028       69580 :   pari_sp av = avma;
    2029      503527 :   while ((p = u_forprime_next(T)) && size/p >= cutoffbits) {
    2030      433983 :     long v = 1;
    2031      433983 :     if (DEBUGLEVEL>5 && cnt++==2000)
    2032           0 :       { cnt=0; err_printf("%lu%% ", 100*p*cutoffbits/size); }
    2033      434037 :     while (is_kth_power(x, p, pt)) {
    2034          56 :       v *= p; x = *pt;
    2035          56 :       size = expi(x);
    2036             :     }
    2037      433989 :     if (v > 1)
    2038             :     {
    2039          42 :       if (DEBUGLEVEL>5) err_printf("\nOddPwrs: is a %ld power\n",v);
    2040          42 :       return v;
    2041             :     }
    2042             :   }
    2043       69537 :   if (DEBUGLEVEL>5) err_printf("\nOddPwrs: not a power\n",p);
    2044       69537 :   return gc_int(av,0); /* give up */
    2045             : }
    2046             : 
    2047             : /***********************************************************************/
    2048             : /**                FACTORIZATION  (master iteration)                  **/
    2049             : /**      Driver for the various methods of finding large factors      **/
    2050             : /**      (after trial division has cast out the very small ones).     **/
    2051             : /**                        GN1998Jun24--30                            **/
    2052             : /***********************************************************************/
    2053             : 
    2054             : /* Direct use:
    2055             :  *  ifac_start_hint(n,moebius,hint) registers with the iterative factorizer
    2056             :  *  - an integer n (without prime factors  < tridiv_bound(n))
    2057             :  *  - registers whether or not we should terminate early if we find a square
    2058             :  *    factor,
    2059             :  *  - a hint about which method(s) to use.
    2060             :  *  This must always be called first. If input is not composite, oo loop.
    2061             :  *  The routine decomposes n nontrivially into a product of two factors except
    2062             :  *  in squarefreeness ('Moebius') mode.
    2063             :  *
    2064             :  *  ifac_start(n,moebius) same using default hint.
    2065             :  *
    2066             :  *  ifac_primary_factor()  returns a prime divisor (not necessarily the
    2067             :  *    smallest) and the corresponding exponent.
    2068             :  *
    2069             :  * Encapsulated user interface: Many arithmetic functions have a 'contributor'
    2070             :  * ifac_xxx, to be called on any large composite cofactor left over after trial
    2071             :  * division by small primes: xxx is one of moebius, issquarefree, totient, etc.
    2072             :  *
    2073             :  * We never test whether the input number is prime or composite, since
    2074             :  * presumably it will have come out of the small factors finder stage
    2075             :  * (which doesn't really exist yet but which will test the left-over
    2076             :  * cofactor for primality once it does). */
    2077             : 
    2078             : /* The data structure in which we preserve whatever we know about our number N
    2079             :  * is kept on the PARI stack, and updated as needed.
    2080             :  * This makes the machinery re-entrant, and avoids memory leaks when a lengthy
    2081             :  * factorization is interrupted. We try to keep the whole affair connected,
    2082             :  * and the parent object is always older than its children.  This may in
    2083             :  * rare cases lead to some extra copying around, and knowing what is garbage
    2084             :  * at any given time is not trivial. See below for examples how to do it right.
    2085             :  * (Connectedness is destroyed if callers of ifac_main() create stuff on the
    2086             :  * stack in between calls. This is harmless as long as ifac_realloc() is used
    2087             :  * to re-create a connected object at the head of the stack just before
    2088             :  * collecting garbage.)
    2089             :  * A t_INT may well have > 10^6 distinct prime factors larger than 2^16. Since
    2090             :  * we need not find factors in order of increasing size, we must be prepared to
    2091             :  * drag a very large amount of data around.  We start with a small structure
    2092             :  * and extend it when necessary. */
    2093             : 
    2094             : /* The idea of the algorithm is:
    2095             :  * Let N0 be whatever is currently left of N after dividing off all the
    2096             :  * prime powers we have already returned to the caller.  Then we maintain
    2097             :  * N0 as a product
    2098             :  * (1) N0 = \prod_i P_i^{e_i} * \prod_j Q_j^{f_j} * \prod_k C_k^{g_k}
    2099             :  * where the P_i and Q_j are distinct primes, each C_k is known composite,
    2100             :  * none of the P_i divides any C_k, and we also know the total ordering
    2101             :  * of all the P_i, Q_j and C_k; in particular, we will never try to divide
    2102             :  * a C_k by a larger Q_j.  Some of the C_k may have common factors.
    2103             :  *
    2104             :  * Caveat implementor:  Taking gcds among C_k's is very likely to cost at
    2105             :  * least as much time as dividing off any primes as we find them, and book-
    2106             :  * keeping would be tough (since D=gcd(C_1,C_2) can still have common factors
    2107             :  * with both C_1/D and C_2/D, and so on...).
    2108             :  *
    2109             :  * At startup, we just initialize the structure to
    2110             :  * (2) N = C_1^1   (composite).
    2111             :  *
    2112             :  * Whenever ifac_primary_factor() or one of the arithmetic user interface
    2113             :  * routines needs a primary factor, and the smallest thing in our list is P_1,
    2114             :  * we return that and its exponent, and remove it from our list. (When nothing
    2115             :  * is left, we return a sentinel value -- gen_1.  And in Moebius mode, when we
    2116             :  * see something with exponent > 1, whether prime or composite, we return gen_0
    2117             :  * or 0, depending on the function). In all other cases, ifac_main() iterates
    2118             :  * the following steps until we have a P_1 in the smallest position.
    2119             :  *
    2120             :  * When the smallest item is C_1, as it is initially:
    2121             :  * (3.1) Crack C_1 into a nontrivial product  U_1 * U_2  by whatever method
    2122             :  * comes to mind for this size. (U for 'unknown'.)  Cracking will detect
    2123             :  * perfect powers, so we may instead see a power of some U_1 here, or even
    2124             :  * something of the form U_1^k*U_2^k; of course the exponent already attached
    2125             :  * to C_1 is taken into account in the following.
    2126             :  * (3.2) If we have U_1*U_2, sort the two factors (distinct: squares are caught
    2127             :  * in stage 3.1). N.B. U_1 and U_2 are smaller than anything else in our list.
    2128             :  * (3.3) Check U_1 and U_2 for primality, and flag them accordingly.
    2129             :  * (3.4) Iterate.
    2130             :  *
    2131             :  * When the smallest item is Q_1:
    2132             :  * This is the unpleasant case.  We go through the entire list and try to
    2133             :  * divide Q_1 off each of the current C_k's, which usually fails, but may
    2134             :  * succeed several times. When a division was successful, the corresponding
    2135             :  * C_k is removed from our list, and the cofactor becomes a U_l for the moment
    2136             :  * unless it is 1 (which happens when C_k was a power of Q_1).  When we're
    2137             :  * through we upgrade Q_1 to P_1 status, then do a primality check on each U_l
    2138             :  * and sort it back into the list either as a Q_j or as a C_k.  If during the
    2139             :  * insertion sort we discover that some U_l equals some P_i or Q_j or C_k we
    2140             :  * already have, we just add U_l's exponent to that of its twin. (The sorting
    2141             :  * therefore happens before the primality test). Since this may produce one or
    2142             :  * more elements smaller than the P_1 we just confirmed, we may have to repeat
    2143             :  * the iteration.
    2144             :  * A trick avoids some Q_1 instances: just after the sweep classifying
    2145             :  * all current unknowns as either composites or primes, we do another downward
    2146             :  * sweep beginning with the largest current factor and stopping just above the
    2147             :  * largest current composite.  Every Q_j we pass is turned into a P_i.
    2148             :  * (Different primes are automatically coprime among each other, and primes do
    2149             :  * not divide smaller composites.)
    2150             :  * NB: We have no use for comparing the square of a prime to N0.  Normally
    2151             :  * we will get called after casting out only the smallest primes, and
    2152             :  * since we cannot guarantee that we see the large prime factors in as-
    2153             :  * cending order, we cannot stop when we find one larger than sqrt(N0). */
    2154             : 
    2155             : /* Data structure: We keep everything in a single t_VEC of t_INTs.  The
    2156             :  * first 2 components are read-only:
    2157             :  * 1) the first records whether we're doing full (NULL) or Moebius (gen_1)
    2158             :  * factorization; in the latter case subroutines return a sentinel value as
    2159             :  * soon as they spot an exponent > 1.
    2160             :  * 2) the second records the hint from factorint()'s optional flag, for use by
    2161             :  * ifac_crack().
    2162             :  *
    2163             :  * The remaining components (initially 15) are used in groups of three:
    2164             :  * [ factor (t_INT), exponent (t_INT), factor class ], where factor class is
    2165             :  *  NULL : unknown
    2166             :  *  gen_0: known composite C_k
    2167             :  *  gen_1: known prime Q_j awaiting trial division
    2168             :  *  gen_2: finished prime P_i.
    2169             :  * When during the division stage we re-sort a C_k-turned-U_l to a lower
    2170             :  * position, we rotate any intervening material upward towards its old
    2171             :  * slot.  When a C_k was divided down to 1, its slot is left empty at
    2172             :  * first; similarly when the re-sorting detects a repeated factor.
    2173             :  * After the sorting phase, we de-fragment the list and squeeze all the
    2174             :  * occupied slots together to the high end, so that ifac_crack() has room
    2175             :  * for new factors.  When this doesn't suffice, we abandon the current vector
    2176             :  * and allocate a somewhat larger one, defragmenting again while copying.
    2177             :  *
    2178             :  * For internal use: note that all exponents will fit into C longs, given
    2179             :  * PARI's lgefint field size.  When we work with them, we sometimes read
    2180             :  * out the GEN pointer, and sometimes do an itos, whatever is more con-
    2181             :  * venient for the task at hand. */
    2182             : 
    2183             : /*** Overview ***/
    2184             : 
    2185             : /* The '*where' argument in the following points into *partial at the first of
    2186             :  * the three fields of the first occupied slot.  It's there because the caller
    2187             :  * would already know where 'here' is, so we don't want to search for it again.
    2188             :  * We do not preserve this from one user-interface call to the next. */
    2189             : 
    2190             : /* In the most cases, control flows from the user interface to ifac_main() and
    2191             :  * then to a succession of ifac_crack()s and ifac_divide()s, with (typically)
    2192             :  * none of the latter finding anything. */
    2193             : 
    2194             : #define LAST(x) x+lg(x)-3
    2195             : #define FIRST(x) x+3
    2196             : 
    2197             : #define MOEBIUS(x) gel(x,1)
    2198             : #define HINT(x) gel(x,2)
    2199             : 
    2200             : /* y <- x */
    2201             : INLINE void
    2202           0 : SHALLOWCOPY(GEN x, GEN y) {
    2203           0 :   VALUE(y) = VALUE(x);
    2204           0 :   EXPON(y) = EXPON(x);
    2205           0 :   CLASS(y) = CLASS(x);
    2206           0 : }
    2207             : /* y <- x */
    2208             : INLINE void
    2209           0 : COPY(GEN x, GEN y) {
    2210           0 :   icopyifstack(VALUE(x), VALUE(y));
    2211           0 :   icopyifstack(EXPON(x), EXPON(y));
    2212           0 :   CLASS(y) = CLASS(x);
    2213           0 : }
    2214             : 
    2215             : /* Diagnostics */
    2216             : static void
    2217           0 : ifac_factor_dbg(GEN x)
    2218             : {
    2219           0 :   GEN c = CLASS(x), v = VALUE(x);
    2220           0 :   if (c == gen_2) err_printf("IFAC: factor %Ps\n\tis prime (finished)\n", v);
    2221           0 :   else if (c == gen_1) err_printf("IFAC: factor %Ps\n\tis prime\n", v);
    2222           0 :   else if (c == gen_0) err_printf("IFAC: factor %Ps\n\tis composite\n", v);
    2223           0 : }
    2224             : static void
    2225           0 : ifac_check(GEN partial, GEN where)
    2226             : {
    2227           0 :   if (!where || where < FIRST(partial) || where > LAST(partial))
    2228           0 :     pari_err_BUG("ifac_check ['where' out of bounds]");
    2229           0 : }
    2230             : static void
    2231           0 : ifac_print(GEN part, GEN where)
    2232             : {
    2233           0 :   long l = lg(part);
    2234             :   GEN p;
    2235             : 
    2236           0 :   err_printf("ifac partial factorization structure: %ld slots, ", (l-3)/3);
    2237           0 :   if (MOEBIUS(part)) err_printf("Moebius mode, ");
    2238           0 :   err_printf("hint = %ld\n", itos(HINT(part)));
    2239           0 :   ifac_check(part, where);
    2240           0 :   for (p = part+3; p < part + l; p += 3)
    2241             :   {
    2242           0 :     GEN v = VALUE(p), e = EXPON(p), c = CLASS(p);
    2243           0 :     const char *s = "";
    2244           0 :     if (!v) { err_printf("[empty slot]\n"); continue; }
    2245           0 :     if (c == NULL) s = "unknown";
    2246           0 :     else if (c == gen_0) s = "composite";
    2247           0 :     else if (c == gen_1) s = "unfinished prime";
    2248           0 :     else if (c == gen_2) s = "prime";
    2249           0 :     else pari_err_BUG("unknown factor class");
    2250           0 :     err_printf("[%Ps, %Ps, %s]\n", v, e, s);
    2251             :   }
    2252           0 :   err_printf("Done.\n");
    2253           0 : }
    2254             : 
    2255             : static const long decomp_default_hint = 0;
    2256             : /* assume n > 0, which we can assign to */
    2257             : /* return initial data structure, see ifac_crack() for the hint argument */
    2258             : static GEN
    2259        5547 : ifac_start_hint(GEN n, int moebius, long hint)
    2260             : {
    2261        5547 :   const long ifac_initial_length = 3 + 7*3;
    2262             :   /* codeword, moebius, hint, 7 slots -- a 512-bit product of distinct 8-bit
    2263             :    * primes needs at most 7 slots at a time) */
    2264        5547 :   GEN here, part = cgetg(ifac_initial_length, t_VEC);
    2265             : 
    2266        5547 :   MOEBIUS(part) = moebius? gen_1 : NULL;
    2267        5547 :   HINT(part) = stoi(hint);
    2268             :   /* fill first slot at the top end */
    2269        5547 :   here = part + ifac_initial_length - 3; /* LAST(part) */
    2270        5547 :   INIT(here, n,gen_1,gen_0); /* n^1: composite */
    2271       38829 :   while ((here -= 3) > part) ifac_delete(here);
    2272        5547 :   return part;
    2273             : }
    2274             : GEN
    2275        1911 : ifac_start(GEN n, int moebius)
    2276        1911 : { return ifac_start_hint(n,moebius,decomp_default_hint); }
    2277             : 
    2278             : /* Return next nonempty slot after 'here', NULL if none exist */
    2279             : static GEN
    2280       15913 : ifac_find(GEN partial)
    2281             : {
    2282       15913 :   GEN scan, end = partial + lg(partial);
    2283             : 
    2284             : #ifdef IFAC_DEBUG
    2285             :   ifac_check(partial, partial);
    2286             : #endif
    2287      116609 :   for (scan = partial+3; scan < end; scan += 3)
    2288      111300 :     if (VALUE(scan)) return scan;
    2289        5309 :   return NULL;
    2290             : }
    2291             : 
    2292             : /* Defragment: squeeze out unoccupied slots above *where. Unoccupied slots
    2293             :  * arise when a composite factor dissolves completely whilst dividing off a
    2294             :  * prime, or when ifac_resort() spots a coincidence and merges two factors.
    2295             :  * Update *where */
    2296             : static void
    2297         224 : ifac_defrag(GEN *partial, GEN *where)
    2298             : {
    2299         224 :   GEN scan_new = LAST(*partial), scan_old;
    2300             : 
    2301         672 :   for (scan_old = scan_new; scan_old >= *where; scan_old -= 3)
    2302             :   {
    2303         448 :     if (!VALUE(scan_old)) continue; /* empty slot */
    2304         448 :     if (scan_old < scan_new) SHALLOWCOPY(scan_old, scan_new);
    2305         448 :     scan_new -= 3; /* point at next slot to be written */
    2306             :   }
    2307         224 :   scan_new += 3; /* back up to last slot written */
    2308         224 :   *where = scan_new;
    2309        1344 :   while ((scan_new -= 3) > *partial) ifac_delete(scan_new); /* erase junk */
    2310         224 : }
    2311             : 
    2312             : /* Move to a larger main vector, updating *where if it points into it, and
    2313             :  * *partial in any case. Can be used as a specialized gcopy before
    2314             :  * a gerepileupto() (pass 0 as the new length). Normally, one would pass
    2315             :  * new_lg=1 to let this function guess the new size.  To be used sparingly.
    2316             :  * Complex version of ifac_defrag(), combined with reallocation.  If new_lg
    2317             :  * is 0, use the old length, so this acts just like gcopy except that the
    2318             :  * 'where' pointer is carried along; if it is 1, we make an educated guess.
    2319             :  * Exception:  If new_lg is 0, the vector is full to the brim, and the first
    2320             :  * entry is composite, we make it longer to avoid being called again a
    2321             :  * microsecond later. It is safe to call this with *where = NULL:
    2322             :  * if it doesn't point anywhere within the old structure, it is left alone */
    2323             : static void
    2324           0 : ifac_realloc(GEN *partial, GEN *where, long new_lg)
    2325             : {
    2326           0 :   long old_lg = lg(*partial);
    2327             :   GEN newpart, scan_new, scan_old;
    2328             : 
    2329           0 :   if (new_lg == 1)
    2330           0 :     new_lg = 2*old_lg - 6;        /* from 7 slots to 13 to 25... */
    2331           0 :   else if (new_lg <= old_lg)        /* includes case new_lg == 0 */
    2332             :   {
    2333           0 :     GEN first = *partial + 3;
    2334           0 :     new_lg = old_lg;
    2335             :     /* structure full and first entry composite or unknown */
    2336           0 :     if (VALUE(first) && (CLASS(first) == gen_0 || CLASS(first)==NULL))
    2337           0 :       new_lg += 6; /* give it a little more breathing space */
    2338             :   }
    2339           0 :   newpart = cgetg(new_lg, t_VEC);
    2340           0 :   if (DEBUGMEM >= 3)
    2341           0 :     err_printf("IFAC: new partial factorization structure (%ld slots)\n",
    2342           0 :                (new_lg - 3)/3);
    2343           0 :   MOEBIUS(newpart) = MOEBIUS(*partial);
    2344           0 :   icopyifstack(HINT(*partial), HINT(newpart));
    2345             :   /* Downward sweep through the old *partial. Pick up 'where' and carry it
    2346             :    * over if we pass it. (Only useful if it pointed at a nonempty slot.)
    2347             :    * Factors are COPY'd so that we again have a nice object (parent older
    2348             :    * than children, connected), except the one factor that may still be living
    2349             :    * in a clone where n originally was; exponents are similarly copied if they
    2350             :    * aren't global constants; class-of-factor fields are global constants so we
    2351             :    * need only copy them as pointers. Caller may then do a gerepileupto() */
    2352           0 :   scan_new = newpart + new_lg - 3; /* LAST(newpart) */
    2353           0 :   scan_old = *partial + old_lg - 3; /* LAST(*partial) */
    2354           0 :   for (; scan_old > *partial + 2; scan_old -= 3)
    2355             :   {
    2356           0 :     if (*where == scan_old) *where = scan_new;
    2357           0 :     if (!VALUE(scan_old)) continue; /* skip empty slots */
    2358           0 :     COPY(scan_old, scan_new); scan_new -= 3;
    2359             :   }
    2360           0 :   scan_new += 3; /* back up to last slot written */
    2361           0 :   while ((scan_new -= 3) > newpart) ifac_delete(scan_new);
    2362           0 :   *partial = newpart;
    2363           0 : }
    2364             : 
    2365             : /* Re-sort one (typically unknown) entry from washere to a new position,
    2366             :  * rotating intervening entries upward to fill the vacant space. If the new
    2367             :  * position is the same as the old one, or the new value of the entry coincides
    2368             :  * with a value already occupying a lower slot, then we just add exponents (and
    2369             :  * use the 'more known' class, and return 1 immediately when in Moebius mode).
    2370             :  * Slots between *where and washere must be in sorted order, so a sweep using
    2371             :  * this to re-sort several unknowns must proceed upward, see ifac_resort().
    2372             :  * Bubble-sort-of-thing sort. Won't be exercised frequently, so this is ok */
    2373             : static void
    2374         112 : ifac_sort_one(GEN *where, GEN washere)
    2375             : {
    2376         112 :   GEN old, scan = washere - 3;
    2377             :   GEN value, exponent, class0, class1;
    2378             :   long cmp_res;
    2379             : 
    2380         112 :   if (scan < *where) return; /* nothing to do, washere==*where */
    2381         112 :   value    = VALUE(washere);
    2382         112 :   exponent = EXPON(washere);
    2383         112 :   class0 = CLASS(washere);
    2384         112 :   cmp_res = -1; /* sentinel */
    2385         112 :   while (scan >= *where) /* at least once */
    2386             :   {
    2387         112 :     if (VALUE(scan))
    2388             :     { /* current slot nonempty, check against where */
    2389         112 :       cmp_res = cmpii(value, VALUE(scan));
    2390         112 :       if (cmp_res >= 0) break; /* have found where to stop */
    2391             :     }
    2392             :     /* copy current slot upward by one position and move pointers down */
    2393           0 :     SHALLOWCOPY(scan, scan+3);
    2394           0 :     scan -= 3;
    2395             :   }
    2396         112 :   scan += 3;
    2397             :   /* At this point there are the following possibilities:
    2398             :    * 1) cmp_res == -1. Either value is less than that at *where, or *where was
    2399             :    * pointing at vacant slots and any factors we saw en route were larger than
    2400             :    * value. At any rate, scan == *where now, and scan is pointing at an empty
    2401             :    * slot, into which we'll stash our entry.
    2402             :    * 2) cmp_res == 0. The entry at scan-3 is the one, we compare class0
    2403             :    * fields and add exponents, and put it all into the vacated scan slot,
    2404             :    * NULLing the one at scan-3 (and possibly updating *where).
    2405             :    * 3) cmp_res == 1. The slot at scan is the one to store our entry into. */
    2406         112 :   if (cmp_res)
    2407             :   {
    2408         112 :     if (cmp_res < 0 && scan != *where)
    2409           0 :       pari_err_BUG("ifact_sort_one [misaligned partial]");
    2410         112 :     INIT(scan, value, exponent, class0); return;
    2411             :   }
    2412             :   /* case cmp_res == 0: repeated factor detected */
    2413           0 :   if (DEBUGLEVEL >= 4)
    2414           0 :     err_printf("IFAC: repeated factor %Ps\n\tin ifac_sort_one\n", value);
    2415           0 :   old = scan - 3;
    2416             :   /* if old class0 was composite and new is prime, or vice versa, complain
    2417             :    * (and if one class0 was unknown and the other wasn't, use the known one) */
    2418           0 :   class1 = CLASS(old);
    2419           0 :   if (class0) /* should never be used */
    2420             :   {
    2421           0 :     if (class1)
    2422             :     {
    2423           0 :       if (class0 == gen_0 && class1 != gen_0)
    2424           0 :         pari_err_BUG("ifac_sort_one (composite = prime)");
    2425           0 :       else if (class0 != gen_0 && class1 == gen_0)
    2426           0 :         pari_err_BUG("ifac_sort_one (prime = composite)");
    2427           0 :       else if (class0 == gen_2)
    2428           0 :         CLASS(scan) = class0;
    2429             :     }
    2430             :     else
    2431           0 :       CLASS(scan) = class0;
    2432             :   }
    2433             :   /* else stay with the existing known class0 */
    2434           0 :   CLASS(scan) = class1;
    2435             :   /* in any case, add exponents */
    2436           0 :   if (EXPON(old) == gen_1 && exponent == gen_1)
    2437           0 :     EXPON(scan) = gen_2;
    2438             :   else
    2439           0 :     EXPON(scan) = addii(EXPON(old), exponent);
    2440             :   /* move the value over and null out the vacated slot below */
    2441           0 :   old = scan - 3;
    2442           0 :   *scan = *old;
    2443           0 :   ifac_delete(old);
    2444             :   /* finally, see whether *where should be pulled in */
    2445           0 :   if (old == *where) *where += 3;
    2446             : }
    2447             : 
    2448             : /* Sort all current unknowns downward to where they belong. Sweeps in the
    2449             :  * upward direction. Not needed after ifac_crack(), only when ifac_divide()
    2450             :  * returned true. Update *where. */
    2451             : static void
    2452         112 : ifac_resort(GEN *partial, GEN *where)
    2453             : {
    2454             :   GEN scan, end;
    2455         112 :   ifac_defrag(partial, where); end = LAST(*partial);
    2456         336 :   for (scan = *where; scan <= end; scan += 3)
    2457         224 :     if (VALUE(scan) && !CLASS(scan)) ifac_sort_one(where, scan); /*unknown*/
    2458         112 :   ifac_defrag(partial, where); /* remove newly created gaps */
    2459         112 : }
    2460             : 
    2461             : /* Let x be a t_INT known not to have small divisors (< 2^14). Return 0 if x
    2462             :  * is a proven composite. Return 1 if we believe it to be prime (fully proven
    2463             :  * prime if factor_proven is set).  */
    2464             : int
    2465       30301 : ifac_isprime(GEN x)
    2466             : {
    2467       30301 :   if (!BPSW_psp_nosmalldiv(x)) return 0; /* composite */
    2468       18787 :   if (factor_proven && ! BPSW_isprime(x))
    2469             :   {
    2470           0 :     pari_warn(warner,
    2471             :               "IFAC: pseudo-prime %Ps\n\tis not prime. PLEASE REPORT!\n", x);
    2472           0 :     return 0;
    2473             :   }
    2474       18787 :   return 1;
    2475             : }
    2476             : 
    2477             : static int
    2478       10671 : ifac_checkprime(GEN x)
    2479             : {
    2480       10671 :   int res = ifac_isprime(VALUE(x));
    2481       10671 :   CLASS(x) = res? gen_1: gen_0;
    2482       10671 :   if (DEBUGLEVEL>2) ifac_factor_dbg(x);
    2483       10671 :   return res;
    2484             : }
    2485             : 
    2486             : /* Determine primality or compositeness of all current unknowns, and set
    2487             :  * class Q primes to finished (class P) if everything larger is already
    2488             :  * known to be prime.  When after_crack >= 0, only look at the
    2489             :  * first after_crack things in the list (do nothing when it's 0) */
    2490             : static void
    2491        5647 : ifac_whoiswho(GEN *partial, GEN *where, long after_crack)
    2492             : {
    2493        5647 :   GEN scan, scan_end = LAST(*partial);
    2494             : 
    2495             : #ifdef IFAC_DEBUG
    2496             :   ifac_check(*partial, *where);
    2497             : #endif
    2498        5647 :   if (after_crack == 0) return;
    2499        4976 :   if (after_crack > 0) /* check at most after_crack entries */
    2500        4864 :     scan = *where + 3*(after_crack - 1); /* assert(scan <= scan_end) */
    2501             :   else
    2502         322 :     for (scan = scan_end; scan >= *where; scan -= 3)
    2503             :     {
    2504         217 :       if (CLASS(scan))
    2505             :       { /* known class of factor */
    2506         105 :         if (CLASS(scan) == gen_0) break;
    2507         105 :         if (CLASS(scan) == gen_1)
    2508             :         {
    2509           0 :           if (DEBUGLEVEL>=3)
    2510             :           {
    2511           0 :             err_printf("IFAC: factor %Ps\n\tis prime (no larger composite)\n",
    2512           0 :                        VALUE(*where));
    2513           0 :             err_printf("IFAC: prime %Ps\n\tappears with exponent = %ld\n",
    2514           0 :                        VALUE(*where), itos(EXPON(*where)));
    2515             :           }
    2516           0 :           CLASS(scan) = gen_2;
    2517             :         }
    2518         105 :         continue;
    2519             :       }
    2520         112 :       if (!ifac_checkprime(scan)) break; /* must disable Q-to-P */
    2521         105 :       CLASS(scan) = gen_2; /* P_i, finished prime */
    2522         105 :       if (DEBUGLEVEL>2) ifac_factor_dbg(scan);
    2523             :     }
    2524             :   /* go on, Q-to-P trick now disabled */
    2525       14802 :   for (; scan >= *where; scan -= 3)
    2526             :   {
    2527        9826 :     if (CLASS(scan)) continue;
    2528        9805 :     (void)ifac_checkprime(scan); /* Qj | Ck */
    2529             :   }
    2530             : }
    2531             : 
    2532             : /* Divide all current composites by first (prime, class Q) entry, updating its
    2533             :  * exponent, and turning it into a finished prime (class P).  Return 1 if any
    2534             :  * such divisions succeeded  (in Moebius mode, the update may then not have
    2535             :  * been completed), or 0 if none of them succeeded.  Doesn't modify *where.
    2536             :  * Here we normally do not check that the first entry is a not-finished
    2537             :  * prime.  Stack management: we may allocate a new exponent */
    2538             : static long
    2539        9877 : ifac_divide(GEN *partial, GEN *where, long moebius_mode)
    2540             : {
    2541        9877 :   GEN scan, scan_end = LAST(*partial);
    2542        9877 :   long res = 0, exponent, newexp, otherexp;
    2543             : 
    2544             : #ifdef IFAC_DEBUG
    2545             :   ifac_check(*partial, *where);
    2546             :   if (CLASS(*where) != gen_1)
    2547             :     pari_err_BUG("ifac_divide [division by composite or finished prime]");
    2548             :   if (!VALUE(*where)) pari_err_BUG("ifac_divide [division by nothing]");
    2549             : #endif
    2550        9877 :   newexp = exponent = itos(EXPON(*where));
    2551        9877 :   if (exponent > 1 && moebius_mode) return 1;
    2552             :   /* should've been caught by caller */
    2553             : 
    2554       14909 :   for (scan = *where+3; scan <= scan_end; scan += 3)
    2555             :   {
    2556        5039 :     if (CLASS(scan) != gen_0) continue; /* the other thing ain't composite */
    2557         431 :     otherexp = 0;
    2558             :     /* divide in place to keep stack clutter minimal */
    2559         550 :     while (dvdiiz(VALUE(scan), VALUE(*where), VALUE(scan)))
    2560             :     {
    2561         126 :       if (moebius_mode) return 1; /* immediately */
    2562         119 :       if (!otherexp) otherexp = itos(EXPON(scan));
    2563         119 :       newexp += otherexp;
    2564             :     }
    2565         424 :     if (newexp > exponent)        /* did anything happen? */
    2566             :     {
    2567         112 :       EXPON(*where) = (newexp == 2 ? gen_2 : utoipos(newexp));
    2568         112 :       exponent = newexp;
    2569         112 :       if (is_pm1((GEN)*scan)) /* factor dissolved completely */
    2570             :       {
    2571           0 :         ifac_delete(scan);
    2572           0 :         if (DEBUGLEVEL >= 4)
    2573           0 :           err_printf("IFAC: a factor was a power of another prime factor\n");
    2574             :       } else {
    2575         112 :         CLASS(scan) = NULL;        /* at any rate it's Unknown now */
    2576         112 :         if (DEBUGLEVEL >= 4)
    2577           0 :           err_printf("IFAC: a factor was divisible by another prime factor,\n"
    2578             :                      "\tleaving a cofactor = %Ps\n", VALUE(scan));
    2579             :       }
    2580         112 :       res = 1;
    2581         112 :       if (DEBUGLEVEL >= 5)
    2582           0 :         err_printf("IFAC: prime %Ps\n\tappears at least to the power %ld\n",
    2583           0 :                    VALUE(*where), newexp);
    2584             :     }
    2585             :   } /* for */
    2586        9870 :   CLASS(*where) = gen_2; /* make it a finished prime */
    2587        9870 :   if (DEBUGLEVEL >= 3)
    2588           0 :     err_printf("IFAC: prime %Ps\n\tappears with exponent = %ld\n",
    2589           0 :                VALUE(*where), newexp);
    2590        9870 :   return res;
    2591             : }
    2592             : 
    2593             : /* found out our integer was factor^exp. Update */
    2594             : static void
    2595         950 : update_pow(GEN where, GEN factor, long exp, pari_sp *av)
    2596             : {
    2597         950 :   GEN ex = EXPON(where);
    2598         950 :   if (DEBUGLEVEL>3)
    2599           0 :     err_printf("IFAC: found %Ps =\n\t%Ps ^%ld\n", *where, factor, exp);
    2600         950 :   affii(factor, VALUE(where)); set_avma(*av);
    2601         950 :   if (ex == gen_1)
    2602         747 :   { EXPON(where) = exp == 2? gen_2: utoipos(exp); *av = avma; }
    2603         203 :   else if (ex == gen_2)
    2604         182 :   { EXPON(where) = utoipos(exp<<1); *av = avma; }
    2605             :   else
    2606          21 :     affsi(exp * itos(ex), EXPON(where));
    2607         950 : }
    2608             : /* hint = 0 : Use a default strategy
    2609             :  * hint & 1 : avoid MPQS
    2610             :  * hint & 2 : avoid first-stage ECM (may fall back to ECM if MPQS gives up)
    2611             :  * hint & 4 : avoid Pollard and SQUFOF stages.
    2612             :  * hint & 8 : avoid final ECM; may flag a composite as prime. */
    2613             : #define get_hint(partial) (itos(HINT(*partial)) & 15)
    2614             : 
    2615             : /* Complete ifac_crack's job when a factoring engine splits the current factor
    2616             :  * into a product of three or more new factors. Makes room for them if
    2617             :  * necessary, sorts them, gives them the right exponents and class. Returns the
    2618             :  * number of factors actually written, which may be less than #facvec if there
    2619             :  * are duplicates. Vectors of factors (cf pollardbrent() or mpqs()) contain
    2620             :  * 'slots' of 3 GENs per factor, interpreted as in our partial factorization
    2621             :  * data structure. Thus engines can tell us what they already know about
    2622             :  * factors being prime/composite or appearing to a power larger than thefirst.
    2623             :  * Don't collect garbage.  No diagnostics: engine has printed what it found.
    2624             :  * facvec contains slots of three components per factor; repeated factors are
    2625             :  * allowed (their classes shouldn't contradict each other whereas their
    2626             :  * exponents will be added up) */
    2627             : static long
    2628        1082 : ifac_insert_multiplet(GEN *partial, GEN *where, GEN facvec, long moebius_mode)
    2629             : {
    2630        1082 :   long j,k=1, lfv=lg(facvec)-1, nf=lfv/3, room=(long)(*where-*partial);
    2631             :   /* one of the factors will go into the *where slot, so room is now 3 times
    2632             :    * the number of slots we can use */
    2633        1082 :   long needroom = lfv - room;
    2634        1082 :   GEN newexp, cur, sorted, auxvec = cgetg(nf+1, t_VEC), factor;
    2635        1082 :   long E = itos(EXPON(*where)); /* the old exponent */
    2636             : 
    2637        1082 :   if (DEBUGLEVEL >= 5) /* squfof may return a single squared factor as a set */
    2638           0 :     err_printf("IFAC: incorporating set of %ld factor(s)\n", nf);
    2639        1082 :   if (needroom > 0) ifac_realloc(partial, where, lg(*partial) + needroom);
    2640             : 
    2641             :   /* create sort permutation from the values of the factors */
    2642        3337 :   for (j=nf; j; j--) gel(auxvec,j) = gel(facvec,3*j-2);
    2643        1082 :   sorted = ZV_indexsort(auxvec);
    2644             :   /* store factors, beginning at *where, and catching any duplicates */
    2645        1082 :   cur = facvec + 3*sorted[nf]-2; /* adjust for triple spacing */
    2646        1082 :   VALUE(*where) = VALUE(cur);
    2647        1082 :   newexp = EXPON(cur);
    2648             :   /* if new exponent is 1, the old exponent already in place will do */
    2649        1082 :   if (newexp != gen_1) EXPON(*where) = mului(E,newexp);
    2650        1082 :   CLASS(*where) = CLASS(cur);
    2651        1082 :   if (DEBUGLEVEL >= 6) err_printf("\tstored (largest) factor no. %ld...\n", nf);
    2652             : 
    2653        2255 :   for (j=nf-1; j; j--)
    2654             :   {
    2655             :     GEN e;
    2656        1173 :     cur = facvec + 3*sorted[j]-2;
    2657        1173 :     factor = VALUE(cur);
    2658        1173 :     if (equalii(factor, VALUE(*where)))
    2659             :     {
    2660           0 :       if (DEBUGLEVEL >= 6)
    2661           0 :         err_printf("\tfactor no. %ld is a duplicate%s\n", j, (j>1? "...": ""));
    2662             :       /* update exponent, ignore class which would already have been set,
    2663             :        * then forget current factor */
    2664           0 :       newexp = EXPON(cur);
    2665           0 :       if (newexp != gen_1) /* new exp > 1 */
    2666           0 :         e = addiu(EXPON(*where), E * itou(newexp));
    2667           0 :       else if (EXPON(*where) == gen_1 && E == 1)
    2668           0 :         e = gen_2;
    2669             :       else
    2670           0 :         e = addiu(EXPON(*where), E);
    2671           0 :       EXPON(*where) = e;
    2672             : 
    2673           0 :       if (moebius_mode) return 0; /* stop now, with exponent updated */
    2674           0 :       continue;
    2675             :     }
    2676             : 
    2677        1173 :     *where -= 3;
    2678        1173 :     CLASS(*where) = CLASS(cur); /* class as given */
    2679        1173 :     newexp = EXPON(cur);
    2680        1173 :     if (newexp != gen_1) /* new exp > 1 */
    2681          35 :       e = (E == 1 && newexp == gen_2)? gen_2: mului(E, newexp);
    2682             :     else /* inherit parent's exponent */
    2683        1138 :       e = (E == 1 ? gen_1 : (E == 2 ? gen_2 : utoipos(E)));
    2684        1173 :     EXPON(*where) = e;
    2685             :     /* keep components younger than *partial */
    2686        1173 :     icopyifstack(factor, VALUE(*where));
    2687        1173 :     k++;
    2688        1173 :     if (DEBUGLEVEL >= 6)
    2689           0 :       err_printf("\tfactor no. %ld was unique%s\n", j, j>1? " (so far)...": "");
    2690             :   }
    2691        1082 :   return k;
    2692             : }
    2693             : 
    2694             : /* Split the first (composite) entry.  There _must_ already be room for another
    2695             :  * factor below *where, and *where is updated. Two cases:
    2696             :  * - entry = factor^k is a pure power: factor^k is inserted, leaving *where
    2697             :  *   unchanged;
    2698             :  * - entry = factor * cofactor (not necessarily coprime): both factors are
    2699             :  *   inserted in the correct order, updating *where
    2700             :  * The inserted factors class is set to unknown, they inherit the exponent
    2701             :  * (or a multiple thereof) of their ancestor.
    2702             :  *
    2703             :  * Returns number of factors written into the structure, normally 2 (1 if pure
    2704             :  * power, maybe > 2 if a factoring engine returned a vector of factors instead
    2705             :  * of a single factor). Can reallocate the data structure in the
    2706             :  * vector-of-factors case, not in the most common single-factor case.
    2707             :  * Stack housekeeping:  this routine may create one or more objects  (a new
    2708             :  * factor, or possibly several, and perhaps one or more new exponents > 2) */
    2709             : static long
    2710        5542 : ifac_crack(GEN *partial, GEN *where, long moebius_mode)
    2711             : {
    2712        5542 :   long cmp_res, hint = get_hint(partial);
    2713             :   GEN factor, exponent;
    2714             : 
    2715             : #ifdef IFAC_DEBUG
    2716             :   ifac_check(*partial, *where);
    2717             :   if (*where < *partial + 6)
    2718             :     pari_err_BUG("ifac_crack ['*where' out of bounds]");
    2719             :   if (!(VALUE(*where)) || typ(VALUE(*where)) != t_INT)
    2720             :     pari_err_BUG("ifac_crack [incorrect VALUE(*where)]");
    2721             :   if (CLASS(*where) != gen_0)
    2722             :     pari_err_BUG("ifac_crack [operand not known composite]");
    2723             : #endif
    2724             : 
    2725        5542 :   if (DEBUGLEVEL>2) {
    2726           0 :     err_printf("IFAC: cracking composite\n\t%Ps\n", **where);
    2727           0 :     if (DEBUGLEVEL>3) err_printf("IFAC: checking for pure square\n");
    2728             :   }
    2729             :   /* MPQS cannot factor prime powers. Look for pure powers even if MPQS is
    2730             :    * blocked by hint: fast and useful in bounded factorization */
    2731             :   {
    2732             :     forprime_t T;
    2733        5542 :     ulong exp = 1, mask = 7;
    2734        5542 :     long good = 0;
    2735        5542 :     pari_sp av = avma;
    2736        5542 :     (void)u_forprime_init(&T, 11, ULONG_MAX);
    2737             :     /* crack squares */
    2738        6443 :     while (Z_issquareall(VALUE(*where), &factor))
    2739             :     {
    2740         908 :       good = 1; /* remember we succeeded once */
    2741         908 :       update_pow(*where, factor, 2, &av);
    2742        1579 :       if (moebius_mode) return 0; /* no need to carry on */
    2743             :     }
    2744        5577 :     while ( (exp = is_357_power(VALUE(*where), &factor, &mask)) )
    2745             :     {
    2746          42 :       good = 1; /* remember we succeeded once */
    2747          42 :       update_pow(*where, factor, exp, &av);
    2748          42 :       if (moebius_mode) return 0; /* no need to carry on */
    2749             :     }
    2750             :     /* cutoff at 14 bits as trial division must have found everything below */
    2751        5535 :     while ( (exp = is_pth_power(VALUE(*where), &factor, &T, 15)) )
    2752             :     {
    2753           0 :       good = 1; /* remember we succeeded once */
    2754           0 :       update_pow(*where, factor, exp, &av);
    2755           0 :       if (moebius_mode) return 0; /* no need to carry on */
    2756             :     }
    2757             : 
    2758        5535 :     if (good && hint != 15 && ifac_checkprime(*where))
    2759             :     { /* our composite was a prime power */
    2760         671 :       if (DEBUGLEVEL>3)
    2761           0 :         err_printf("IFAC: factor %Ps\n\tis prime\n", VALUE(*where));
    2762         671 :       return 0; /* bypass subsequent ifac_whoiswho() call */
    2763             :     }
    2764             :   } /* pure power stage */
    2765             : 
    2766        4864 :   factor = NULL;
    2767        4864 :   if (!(hint & 4))
    2768             :   { /* pollardbrent() Rho usually gets a first chance */
    2769        4815 :     if (DEBUGLEVEL >= 4) err_printf("IFAC: trying Pollard-Brent rho method\n");
    2770        4815 :     factor = pollardbrent(VALUE(*where));
    2771        4815 :     if (!factor)
    2772             :     { /* Shanks' squfof() */
    2773        3152 :       if (DEBUGLEVEL >= 4)
    2774           0 :         err_printf("IFAC: trying Shanks' SQUFOF, will fail silently if input\n"
    2775             :                    "      is too large for it.\n");
    2776        3152 :       factor = squfof(VALUE(*where));
    2777             :     }
    2778             :   }
    2779        4864 :   if (!factor && !(hint & 2))
    2780             :   { /* First ECM stage */
    2781         939 :     if (DEBUGLEVEL >= 4) err_printf("IFAC: trying Lenstra-Montgomery ECM\n");
    2782         939 :     factor = ellfacteur(VALUE(*where), 0); /* do not insist */
    2783             :   }
    2784        4864 :   if (!factor && !(hint & 1))
    2785             :   { /* MPQS stage */
    2786         963 :     if (DEBUGLEVEL >= 4) err_printf("IFAC: trying MPQS\n");
    2787         963 :     factor = mpqs(VALUE(*where));
    2788             :   }
    2789        4864 :   if (!factor)
    2790             :   {
    2791          14 :     if (!(hint & 8))
    2792             :     { /* still no luck? Final ECM stage, guaranteed to succeed */
    2793           7 :       if (DEBUGLEVEL >= 4)
    2794           0 :         err_printf("IFAC: forcing ECM, may take some time\n");
    2795           7 :       factor = ellfacteur(VALUE(*where), 1);
    2796             :     }
    2797             :     else
    2798             :     { /* limited factorization */
    2799           7 :       if (DEBUGLEVEL >= 2)
    2800             :       {
    2801           0 :         if (hint != 15)
    2802           0 :           pari_warn(warner, "IFAC: unfactored composite declared prime");
    2803             :         else
    2804           0 :           pari_warn(warner, "IFAC: untested integer declared prime");
    2805             : 
    2806             :         /* don't print it out at level 3 or above, where it would appear
    2807             :          * several times before and after this message already */
    2808           0 :         if (DEBUGLEVEL == 2) err_printf("\t%Ps\n", VALUE(*where));
    2809             :       }
    2810           7 :       CLASS(*where) = gen_1; /* might as well trial-divide by it... */
    2811           7 :       return 1;
    2812             :     }
    2813             :   }
    2814        4857 :   if (typ(factor) == t_VEC) /* delegate this case */
    2815        1082 :     return ifac_insert_multiplet(partial, where, factor, moebius_mode);
    2816             :   /* typ(factor) == t_INT */
    2817             :   /* got single integer back:  work out the cofactor (in place) */
    2818        3775 :   if (!dvdiiz(VALUE(*where), factor, VALUE(*where)))
    2819             :   {
    2820           0 :     err_printf("IFAC: factoring %Ps\n", VALUE(*where));
    2821           0 :     err_printf("\tyielded 'factor' %Ps\n\twhich isn't!\n", factor);
    2822           0 :     pari_err_BUG("factoring");
    2823             :   }
    2824             :   /* factoring engines report the factor found; tell about the cofactor */
    2825        3775 :   if (DEBUGLEVEL >= 4) err_printf("IFAC: cofactor = %Ps\n", VALUE(*where));
    2826             : 
    2827             :   /* The two factors are 'factor' and VALUE(*where), find out which is larger */
    2828        3775 :   cmp_res = cmpii(factor, VALUE(*where));
    2829        3775 :   CLASS(*where) = NULL; /* mark factor /cofactor 'unknown' */
    2830        3775 :   exponent = EXPON(*where);
    2831        3775 :   *where -= 3;
    2832        3775 :   CLASS(*where) = NULL; /* mark factor /cofactor 'unknown' */
    2833        3775 :   icopyifstack(exponent, EXPON(*where));
    2834        3775 :   if (cmp_res < 0)
    2835        3460 :     VALUE(*where) = factor; /* common case */
    2836         315 :   else if (cmp_res > 0)
    2837             :   { /* factor > cofactor, rearrange */
    2838         315 :     GEN old = *where + 3;
    2839         315 :     VALUE(*where) = VALUE(old); /* move cofactor pointer to lowest slot */
    2840         315 :     VALUE(old) = factor; /* save factor */
    2841             :   }
    2842           0 :   else pari_err_BUG("ifac_crack [Z_issquareall miss]");
    2843        3775 :   return 2;
    2844             : }
    2845             : 
    2846             : /* main loop:  iterate until smallest entry is a finished prime;  returns
    2847             :  * a 'where' pointer, or NULL if nothing left, or gen_0 in Moebius mode if
    2848             :  * we aren't squarefree */
    2849             : static GEN
    2850       14959 : ifac_main(GEN *partial)
    2851             : {
    2852       14959 :   const long moebius_mode = !!MOEBIUS(*partial);
    2853       14959 :   GEN here = ifac_find(*partial);
    2854             :   long nf;
    2855             : 
    2856       14959 :   if (!here) return NULL; /* nothing left */
    2857             :   /* loop until first entry is a finished prime.  May involve reallocations,
    2858             :    * thus updates of *partial */
    2859       25373 :   while (CLASS(here) != gen_2)
    2860             :   {
    2861       15419 :     if (CLASS(here) == gen_0) /* composite: crack it */
    2862             :     { /* make sure there's room for another factor */
    2863        5542 :       if (here < *partial + 6)
    2864             :       {
    2865           0 :         ifac_defrag(partial, &here);
    2866           0 :         if (here < *partial + 6) ifac_realloc(partial, &here, 1); /* no luck */
    2867             :       }
    2868        5542 :       nf = ifac_crack(partial, &here, moebius_mode);
    2869        5542 :       if (moebius_mode && EXPON(here) != gen_1) /* that was a power */
    2870             :       {
    2871           7 :         if (DEBUGLEVEL >= 3)
    2872           0 :           err_printf("IFAC: main loop: repeated new factor\n\t%Ps\n", *here);
    2873           7 :         return gen_0;
    2874             :       }
    2875             :       /* deal with the new unknowns.  No sort: ifac_crack did it */
    2876        5535 :       ifac_whoiswho(partial, &here, nf);
    2877        5535 :       continue;
    2878             :     }
    2879        9877 :     if (CLASS(here) == gen_1) /* prime but not yet finished: finish it */
    2880             :     {
    2881        9877 :       if (ifac_divide(partial, &here, moebius_mode))
    2882             :       {
    2883         119 :         if (moebius_mode)
    2884             :         {
    2885           7 :           if (DEBUGLEVEL >= 3)
    2886           0 :             err_printf("IFAC: main loop: another factor was divisible by\n"
    2887             :                        "\t%Ps\n", *here);
    2888           7 :           return gen_0;
    2889             :         }
    2890         112 :         ifac_resort(partial, &here); /* sort new cofactors down */
    2891         112 :         ifac_whoiswho(partial, &here, -1);
    2892             :       }
    2893        9870 :       continue;
    2894             :     }
    2895           0 :     pari_err_BUG("ifac_main [nonexistent factor class]");
    2896             :   } /* while */
    2897        9954 :   if (moebius_mode && EXPON(here) != gen_1)
    2898             :   {
    2899           0 :     if (DEBUGLEVEL >= 3)
    2900           0 :       err_printf("IFAC: after main loop: repeated old factor\n\t%Ps\n", *here);
    2901           0 :     return gen_0;
    2902             :   }
    2903        9954 :   if (DEBUGLEVEL >= 4)
    2904             :   {
    2905           0 :     nf = (*partial + lg(*partial) - here - 3)/3;
    2906           0 :     if (nf)
    2907           0 :       err_printf("IFAC: main loop: %ld factor%s left\n", nf, (nf>1)? "s": "");
    2908             :     else
    2909           0 :       err_printf("IFAC: main loop: this was the last factor\n");
    2910             :   }
    2911        9954 :   if (factor_add_primes && !(get_hint(partial) & 8))
    2912             :   {
    2913           0 :     GEN p = VALUE(here);
    2914           0 :     if (lgefint(p)>3 || uel(p,2) > 0x1000000UL) (void)addprimes(p);
    2915             :   }
    2916        9954 :   return here;
    2917             : }
    2918             : 
    2919             : /* Encapsulated routines */
    2920             : 
    2921             : /* prime/exponent pairs need to appear contiguously on the stack, but we also
    2922             :  * need our data structure somewhere, and we don't know in advance how many
    2923             :  * primes will turn up.  The following discipline achieves this:  When
    2924             :  * ifac_decomp() is called, n should point at an object older than the oldest
    2925             :  * small prime/exponent pair  (ifactor() guarantees this).
    2926             :  * We allocate sufficient space to accommodate several pairs -- eleven pairs
    2927             :  * ought to fit in a space not much larger than n itself -- before calling
    2928             :  * ifac_start().  If we manage to complete the factorization before we run out
    2929             :  * of space, we free the data structure and cull the excess reserved space
    2930             :  * before returning.  When we do run out, we have to leapfrog to generate more
    2931             :  * (guesstimating the requirements from what is left in the partial
    2932             :  * factorization structure);  room for fresh pairs is allocated at the head of
    2933             :  * the stack, followed by an ifac_realloc() to reconnect the data structure and
    2934             :  * move it out of the way, followed by a few pointer tweaks to connect the new
    2935             :  * pairs space to the old one. This whole affair translates into a surprisingly
    2936             :  * compact routine. */
    2937             : 
    2938             : /* find primary factors of n; destroy n */
    2939             : static long
    2940        2643 : ifac_decomp(GEN n, long hint)
    2941             : {
    2942        2643 :   pari_sp av = avma;
    2943        2643 :   long nb = 0;
    2944        2643 :   GEN part, here, workspc, pairs = (GEN)av;
    2945             : 
    2946             :   /* workspc will be doled out in pairs of smaller t_INTs. For n = prod p^{e_p}
    2947             :    * (p not necessarily prime), need room to store all p and e_p [ cgeti(3) ],
    2948             :    * bounded by
    2949             :    *    sum_{p | n} ( log_{2^BIL} (p) + 6 ) <= log_{2^BIL} n + 6 log_2 n */
    2950        2643 :   workspc = new_chunk((expi(n) + 1) * 7);
    2951        2643 :   part = ifac_start_hint(n, 0, hint);
    2952             :   for (;;)
    2953             :   {
    2954        7765 :     here = ifac_main(&part);
    2955        7765 :     if (!here) break;
    2956        5122 :     if (gc_needed(av,1))
    2957             :     {
    2958             :       long offset;
    2959           0 :       if(DEBUGMEM>1)
    2960             :       {
    2961           0 :         pari_warn(warnmem,"[2] ifac_decomp");
    2962           0 :         ifac_print(part, here);
    2963             :       }
    2964           0 :       ifac_realloc(&part, &here, 0);
    2965           0 :       offset = here - part;
    2966           0 :       part = gerepileupto((pari_sp)workspc, part);
    2967           0 :       here = part + offset;
    2968             :     }
    2969        5122 :     nb++;
    2970        5122 :     pairs = icopy_avma(VALUE(here), (pari_sp)pairs);
    2971        5122 :     pairs = icopy_avma(EXPON(here), (pari_sp)pairs);
    2972        5122 :     ifac_delete(here);
    2973             :   }
    2974        2643 :   set_avma((pari_sp)pairs);
    2975        2643 :   if (DEBUGLEVEL >= 3)
    2976           0 :     err_printf("IFAC: found %ld large prime (power) factor%s.\n",
    2977             :                nb, (nb>1? "s": ""));
    2978        2643 :   return nb;
    2979             : }
    2980             : 
    2981             : /***********************************************************************/
    2982             : /**            ARITHMETIC FUNCTIONS WITH EARLY-ABORT                  **/
    2983             : /**  needing direct access to the factoring machinery to avoid work:  **/
    2984             : /**  e.g. if we find a square factor, moebius returns 0, core doesn't **/
    2985             : /**  need to factor it, etc.                                          **/
    2986             : /***********************************************************************/
    2987             : /* memory management */
    2988             : static void
    2989           0 : ifac_GC(pari_sp av, GEN *part)
    2990             : {
    2991           0 :   GEN here = NULL;
    2992           0 :   if(DEBUGMEM>1) pari_warn(warnmem,"ifac_xxx");
    2993           0 :   ifac_realloc(part, &here, 0);
    2994           0 :   *part = gerepileupto(av, *part);
    2995           0 : }
    2996             : 
    2997             : /* destroys n */
    2998             : static long
    2999         211 : ifac_moebius(GEN n)
    3000             : {
    3001         211 :   long mu = 1;
    3002         211 :   pari_sp av = avma;
    3003         211 :   GEN part = ifac_start(n, 1);
    3004             :   for(;;)
    3005         394 :   {
    3006             :     long v;
    3007             :     GEN p;
    3008         605 :     if (!ifac_next(&part,&p,&v)) return v? 0: mu;
    3009         394 :     mu = -mu;
    3010         394 :     if (gc_needed(av,1)) ifac_GC(av,&part);
    3011             :   }
    3012             : }
    3013             : 
    3014             : int
    3015         658 : ifac_read(GEN part, GEN *p, long *e)
    3016             : {
    3017         658 :   GEN here = ifac_find(part);
    3018         658 :   if (!here) return 0;
    3019         340 :   *p = VALUE(here);
    3020         340 :   *e = EXPON(here)[2];
    3021         340 :   return 1;
    3022             : }
    3023             : void
    3024         296 : ifac_skip(GEN part)
    3025             : {
    3026         296 :   GEN here = ifac_find(part);
    3027         296 :   if (here) ifac_delete(here);
    3028         296 : }
    3029             : 
    3030             : /* destroys n */
    3031             : static int
    3032           7 : ifac_ispowerful(GEN n)
    3033             : {
    3034           7 :   pari_sp av = avma;
    3035           7 :   GEN part = ifac_start(n, 0);
    3036             :   for(;;)
    3037           7 :   {
    3038             :     long e;
    3039             :     GEN p;
    3040          14 :     if (!ifac_read(part,&p,&e)) return 1;
    3041             :     /* power: skip */
    3042           7 :     if (e != 1 || Z_isanypower(p,NULL)) { ifac_skip(part); continue; }
    3043           0 :     if (!ifac_next(&part,&p,&e)) return 1;
    3044           0 :     if (e == 1) return 0;
    3045           0 :     if (gc_needed(av,1)) ifac_GC(av,&part);
    3046             :   }
    3047             : }
    3048             : /* destroys n */
    3049             : static GEN
    3050         311 : ifac_core(GEN n)
    3051             : {
    3052         311 :   GEN m = gen_1, c = cgeti(lgefint(n));
    3053         311 :   pari_sp av = avma;
    3054         311 :   GEN part = ifac_start(n, 0);
    3055             :   for(;;)
    3056         333 :   {
    3057             :     long e;
    3058             :     GEN p;
    3059         644 :     if (!ifac_read(part,&p,&e)) return m;
    3060             :     /* square: skip */
    3061         333 :     if (!odd(e) || Z_issquare(p)) { ifac_skip(part); continue; }
    3062          44 :     if (!ifac_next(&part,&p,&e)) return m;
    3063          44 :     if (odd(e)) m = mulii(m, p);
    3064          44 :     if (gc_needed(av,1)) { affii(m,c); m=c; ifac_GC(av,&part); }
    3065             :   }
    3066             : }
    3067             : 
    3068             : /* Where to stop trial dividing in factorization. Guaranteed >= 2^14 */
    3069             : ulong
    3070     1006869 : tridiv_bound(GEN n)
    3071             : {
    3072     1006869 :   ulong l = (ulong)expi(n) + 1;
    3073     1006877 :   if (l <= 32)  return 1UL<<14;
    3074      213014 :   if (l <= 512) return (l-16) << 10;
    3075        1204 :   return 1UL<<19; /* Rho is generally faster above this */
    3076             : }
    3077             : 
    3078             : /* return a value <= (48 << 10) = 49152 < primelinit */
    3079             : ulong
    3080    28273705 : tridiv_boundu(ulong n)
    3081             : {
    3082             : #ifdef LONG_IS_64BIT
    3083    24395696 :   if (n & HIGHMASK)
    3084      771116 :     return ((ulong)expu(n) + 1 - 16) << 10;
    3085             : #else
    3086             :   (void)n;
    3087             : #endif
    3088    27502589 :   return 1UL<<14;
    3089             : }
    3090             : 
    3091             : /* destroys n */
    3092             : static void
    3093         993 : ifac_factoru(GEN n, long hint, GEN P, GEN E, long *pi)
    3094             : {
    3095         993 :   GEN part = ifac_start_hint(n, 0, hint);
    3096             :   for(;;)
    3097        1896 :   {
    3098             :     long v;
    3099             :     GEN p;
    3100        2889 :     if (!ifac_next(&part,&p,&v)) return;
    3101        1896 :     P[*pi] = itou(p);
    3102        1896 :     E[*pi] = v;
    3103        1896 :     (*pi)++;
    3104             :   }
    3105             : }
    3106             : /* destroys n */
    3107             : static long
    3108        1074 : ifac_moebiusu(GEN n)
    3109             : {
    3110        1074 :   GEN part = ifac_start(n, 1);
    3111        1074 :   long s = 1;
    3112             :   for(;;)
    3113        2148 :   {
    3114             :     long v;
    3115             :     GEN p;
    3116        3222 :     if (!ifac_next(&part,&p,&v)) return v? 0: s;
    3117        2148 :     s = -s;
    3118             :   }
    3119             : }
    3120             : 
    3121             : INLINE ulong
    3122   234455933 : u_forprime_next_fast(forprime_t *T)
    3123             : {
    3124   234455933 :   if (*(T->d))
    3125             :   {
    3126   234465392 :     NEXT_PRIME_VIADIFF(T->p, T->d);
    3127   234465392 :     return T->p > T->b ? 0: T->p;
    3128             :   }
    3129           0 :   return u_forprime_next(T);
    3130             : }
    3131             : 
    3132             : /* uisprime(n) knowing n has no prime divisor
    3133             :  *    < all if all != 0,
    3134             :  *    < tridiv_boundu(n) >= 2^14 if all = 0 */
    3135             : static int
    3136       52217 : uisprime_nosmall(ulong n, ulong all)
    3137             : {
    3138       52217 :   return (!all || all > 661) ? uisprime_661(n): uisprime(n);
    3139             : }
    3140             : /* Factor n and output [p,e] where
    3141             :  * p, e are vecsmall with n = prod{p[i]^e[i]}. If all != 0:
    3142             :  * if pU1 is not NULL, set *pU1 and *pU2 so that unfactored composite is
    3143             :  * U1^U2 with U1 not a pure power; else include it in factorization */
    3144             : static GEN
    3145    32078344 : factoru_sign(ulong n, ulong all, long hint, ulong *pU1, ulong *pU2)
    3146             : {
    3147             :   GEN f, E, E2, P, P2;
    3148             :   pari_sp av;
    3149             :   ulong p;
    3150    32078344 :   long i, oldi = -1;
    3151             :   forprime_t S;
    3152             : 
    3153    32078344 :   if (pU1) *pU1 = *pU2 = 1;
    3154    32078344 :   if (n == 0) retmkvec2(mkvecsmall(0), mkvecsmall(1));
    3155    32078344 :   if (n == 1) retmkvec2(cgetg(1,t_VECSMALL), cgetg(1,t_VECSMALL));
    3156             : 
    3157    31645194 :   f = cgetg(3,t_VEC); av = avma;
    3158             :   /* enough room to store <= 15 primes and exponents (OK if n < 2^64) */
    3159    31642878 :   (void)new_chunk(16*2);
    3160    31641916 :   P = cgetg(16, t_VECSMALL); i = 1;
    3161    31640947 :   E = cgetg(16, t_VECSMALL);
    3162    31640491 :   if (!all || all > 2)
    3163             :   {
    3164             :     ulong lim;
    3165    31640424 :     long v = vals(n);
    3166    31640571 :     if (v)
    3167             :     {
    3168    11844688 :       P[1] = 2; E[1] = v; i = 2;
    3169    11844688 :       n >>= v; if (n == 1) goto END;
    3170             :     }
    3171    28472994 :     lim = all? all-1: tridiv_boundu(n);
    3172    28473979 :     if (!(hint & 16) && lim >= 128) /* expu(lim) >= 7 */
    3173      147719 :     { /* fast trial division */
    3174    14165423 :       GEN PR = prodprimes();
    3175    14165540 :       long nPR = lg(PR)-1, b = minss(nPR, expu(lim)-6);
    3176    14165522 :       ulong nr = ugcd(n, umodiu(gel(PR,b), n));
    3177    14167002 :       if (nr != 1)
    3178             :       {
    3179    14098797 :         GEN F = factoru_sign(nr, all, 1 + 2 + 16, NULL, NULL), Q = gel(F,1);
    3180    14097511 :         long j, l = lg(Q);
    3181    38085170 :         for (j = 1; j < l; j++)
    3182             :         {
    3183    23986705 :           ulong p = uel(Q,j);
    3184    23986705 :           if (all && p >= all) break; /* may occur for last p */
    3185    23986699 :           E[i] = u_lvalrem(n, p, &n); /* > 0 */
    3186    23987659 :           P[i] = p; i++;
    3187             :         }
    3188    14098471 :         if (n == 1) goto END;
    3189             :       }
    3190             :     }
    3191             :     else
    3192             :     {
    3193    14308556 :       u_forprime_init(&S, 3, lim);
    3194   106438203 :       while ( (p = u_forprime_next_fast(&S)) )
    3195             :       {
    3196             :         int stop;
    3197             :         /* tiny integers without small factors are often primes */
    3198   106438423 :         if (p == 673)
    3199             :         {
    3200    14413514 :           if (uisprime_661(n)) { P[i] = n; E[i] = 1; i++; goto END; }
    3201      101228 :           oldi = i;
    3202             :         }
    3203   106438003 :         v = u_lvalrem_stop(&n, p, &stop);
    3204   106438073 :         if (v) {
    3205    12653564 :           P[i] = p;
    3206    12653564 :           E[i] = v; i++;
    3207             :         }
    3208   106438073 :         if (stop) {
    3209    14311866 :           if (n != 1) { P[i] = n; E[i] = 1; i++; }
    3210    14311866 :           goto END;
    3211             :         }
    3212             :       }
    3213             :     }
    3214             :   }
    3215             :   /* if i > oldi (includes oldi = -1) we don't know that n is composite */
    3216      148138 :   if (all)
    3217             :   { /* smallfact: look for easy pure powers then stop */
    3218             : #ifdef LONG_IS_64BIT
    3219       82344 :     ulong mask = all > 563 ? (all > 7129 ? 1: 3): 7;
    3220             : #else
    3221       13544 :     ulong mask = all > 22 ? (all > 83 ? 1: 3): 7;
    3222             : #endif
    3223       95888 :     long k = 1, ex;
    3224       96298 :     while (uissquareall(n, &n)) k <<= 1;
    3225       95901 :     while ( (ex = uis_357_power(n, &n, &mask)) ) k *= ex;
    3226       95888 :     if (pU1 && (i == oldi || !uisprime(n)))
    3227         260 :     { *pU1 = n; *pU2 = (ulong)k; }
    3228             :     else
    3229       95628 :     { P[i] = n; E[i] = k; i++; }
    3230       95888 :     goto END;
    3231             :   }
    3232             :   /* we don't known that n is composite ? */
    3233       52250 :   if (oldi != i && uisprime_nosmall(n,all)) { P[i]=n; E[i]=1; i++; goto END; }
    3234             : 
    3235             :   {
    3236             :     GEN perm;
    3237         993 :     ifac_factoru(utoipos(n), hint, P, E, &i);
    3238         993 :     setlg(P, i);
    3239         993 :     perm = vecsmall_indexsort(P);
    3240         993 :     P = vecsmallpermute(P, perm);
    3241         993 :     E = vecsmallpermute(E, perm);
    3242             :   }
    3243    31646958 : END:
    3244    31646958 :   set_avma(av);
    3245    31645479 :   P2 = cgetg(i, t_VECSMALL); gel(f,1) = P2;
    3246    31643120 :   E2 = cgetg(i, t_VECSMALL); gel(f,2) = E2;
    3247    92240206 :   while (--i >= 1) { P2[i] = P[i]; E2[i] = E[i]; }
    3248    31642143 :   return f;
    3249             : }
    3250             : GEN
    3251     3778485 : factoru(ulong n)
    3252     3778485 : { return factoru_sign(n, 0, decomp_default_hint, NULL, NULL); }
    3253             : 
    3254             : ulong
    3255           0 : radicalu(ulong n)
    3256             : {
    3257           0 :   pari_sp av = avma;
    3258           0 :   return gc_long(av, zv_prod(gel(factoru(n),1)));
    3259             : }
    3260             : 
    3261             : long
    3262       54173 : moebiusu_fact(GEN f)
    3263             : {
    3264       54173 :   GEN E = gel(f,2);
    3265       54173 :   long i, l = lg(E);
    3266       93541 :   for (i = 1; i < l; i++)
    3267       57813 :     if (E[i] > 1) return 0;
    3268       35728 :   return odd(l)? 1: -1;
    3269             : }
    3270             : 
    3271             : long
    3272     2513297 : moebiusu(ulong n)
    3273             : {
    3274             :   pari_sp av;
    3275             :   ulong p;
    3276             :   long s, v, test_prime;
    3277             :   forprime_t S;
    3278             : 
    3279     2513297 :   switch(n)
    3280             :   {
    3281           0 :     case 0: (void)check_arith_non0(gen_0,"moebius");/*error*/
    3282      569943 :     case 1: return  1;
    3283      106642 :     case 2: return -1;
    3284             :   }
    3285     1853234 :   v = vals(n);
    3286     1849210 :   if (v == 0)
    3287     1736957 :     s = 1;
    3288             :   else
    3289             :   {
    3290      112253 :     if (v > 1) return 0;
    3291       24253 :     n >>= 1;
    3292       24253 :     s = -1;
    3293             :   }
    3294     1761210 :   av = avma;
    3295     1761210 :   u_forprime_init(&S, 3, tridiv_boundu(n));
    3296     1772550 :   test_prime = 0;
    3297    31241428 :   while ((p = u_forprime_next_fast(&S)))
    3298             :   {
    3299             :     int stop;
    3300             :     /* tiny integers without small factors are often primes */
    3301    31246828 :     if (p == 673)
    3302             :     {
    3303        3752 :       test_prime = 0;
    3304     1781075 :       if (uisprime_661(n)) return gc_long(av,-s);
    3305             :     }
    3306    31245380 :     v = u_lvalrem_stop(&n, p, &stop);
    3307    31243189 :     if (v) {
    3308     1323562 :       if (v > 1) return gc_long(av,0);
    3309     1141607 :       test_prime = 1;
    3310     1141607 :       s = -s;
    3311             :     }
    3312    31061234 :     if (stop) return gc_long(av, n==1? s: -s);
    3313             :   }
    3314        1270 :   set_avma(av);
    3315        1524 :   if (test_prime && uisprime_661(n)) return -s;
    3316             :   else
    3317             :   {
    3318        1074 :     long t = ifac_moebiusu(utoipos(n));
    3319        1074 :     set_avma(av);
    3320        1074 :     if (t == 0) return 0;
    3321        1074 :     return (s == t)? 1: -1;
    3322             :   }
    3323             : }
    3324             : 
    3325             : long
    3326       55374 : moebius(GEN n)
    3327             : {
    3328       55374 :   pari_sp av = avma;
    3329             :   GEN F;
    3330             :   ulong p;
    3331             :   long i, l, s, v;
    3332             :   forprime_t S;
    3333             : 
    3334       55374 :   if ((F = check_arith_non0(n,"moebius")))
    3335             :   {
    3336             :     GEN E;
    3337         784 :     F = clean_Z_factor(F);
    3338         728 :     E = gel(F,2);
    3339         728 :     l = lg(E);
    3340        1428 :     for(i = 1; i < l; i++)
    3341         980 :       if (!equali1(gel(E,i))) return gc_long(av,0);
    3342         448 :     return gc_long(av, odd(l)? 1: -1);
    3343             :   }
    3344       54744 :   if (lgefint(n) == 3) return moebiusu(uel(n,2));
    3345         808 :   p = mod4(n); if (!p) return 0;
    3346         808 :   if (p == 2) { s = -1; n = shifti(n, -1); } else { s = 1; n = icopy(n); }
    3347         808 :   setabssign(n);
    3348             : 
    3349         808 :   u_forprime_init(&S, 3, tridiv_bound(n));
    3350     2322026 :   while ((p = u_forprime_next_fast(&S)))
    3351             :   {
    3352             :     int stop;
    3353     2321446 :     v = Z_lvalrem_stop(&n, p, &stop);
    3354     2321446 :     if (v)
    3355             :     {
    3356        1540 :       if (v > 1) return gc_long(av,0);
    3357        1312 :       s = -s;
    3358        1312 :       if (stop) return gc_long(av, is_pm1(n)? s: -s);
    3359             :     }
    3360             :   }
    3361         580 :   l = lg(primetab);
    3362         590 :   for (i = 1; i < l; i++)
    3363             :   {
    3364          25 :     v = Z_pvalrem(n, gel(primetab,i), &n);
    3365          25 :     if (v)
    3366             :     {
    3367          25 :       if (v > 1) return gc_long(av,0);
    3368          11 :       s = -s;
    3369          11 :       if (is_pm1(n)) return gc_long(av,s);
    3370             :     }
    3371             :   }
    3372         565 :   if (ifac_isprime(n)) return gc_long(av,-s);
    3373             :   /* large composite without small factors */
    3374         211 :   v = ifac_moebius(n);
    3375         211 :   return gc_long(av, s < 0? -v: v); /* correct also if v==0 */
    3376             : }
    3377             : 
    3378             : long
    3379        1708 : ispowerful(GEN n)
    3380             : {
    3381        1708 :   pari_sp av = avma;
    3382             :   GEN F;
    3383             :   ulong p, bound;
    3384             :   long i, l, v;
    3385             :   forprime_t S;
    3386             : 
    3387        1708 :   if ((F = check_arith_all(n, "ispowerful")))
    3388             :   {
    3389         742 :     GEN p, P = gel(F,1), E = gel(F,2);
    3390         742 :     if (lg(P) == 1) return 1; /* 1 */
    3391         728 :     p = gel(P,1);
    3392         728 :     if (!signe(p)) return 1; /* 0 */
    3393         707 :     i = is_pm1(p)? 2: 1; /* skip -1 */
    3394         707 :     l = lg(E);
    3395         980 :     for (; i < l; i++)
    3396         847 :       if (equali1(gel(E,i))) return 0;
    3397         133 :     return 1;
    3398             :   }
    3399         966 :   if (!signe(n)) return 1;
    3400             : 
    3401         952 :   if (mod4(n) == 2) return 0;
    3402         623 :   n = shifti(n, -vali(n));
    3403         623 :   if (is_pm1(n)) return 1;
    3404         546 :   setabssign(n);
    3405         546 :   bound = tridiv_bound(n);
    3406         546 :   u_forprime_init(&S, 3, bound);
    3407      307797 :   while ((p = u_forprime_next_fast(&S)))
    3408             :   {
    3409             :     int stop;
    3410      307790 :     v = Z_lvalrem_stop(&n, p, &stop);
    3411      307790 :     if (v)
    3412             :     {
    3413         742 :       if (v == 1) return gc_long(av,0);
    3414         203 :       if (stop) return gc_long(av, is_pm1(n));
    3415             :     }
    3416             :   }
    3417           7 :   l = lg(primetab);
    3418           7 :   for (i = 1; i < l; i++)
    3419             :   {
    3420           0 :     v = Z_pvalrem(n, gel(primetab,i), &n);
    3421           0 :     if (v)
    3422             :     {
    3423           0 :       if (v == 1) return gc_long(av,0);
    3424           0 :       if (is_pm1(n)) return gc_long(av,1);
    3425             :     }
    3426             :   }
    3427             :   /* no need to factor: must be p^2 or not powerful */
    3428           7 :   if (cmpii(powuu(bound+1, 3), n) > 0) return gc_long(av,  Z_issquare(n));
    3429             : 
    3430           7 :   if (ifac_isprime(n)) return gc_long(av,0);
    3431             :   /* large composite without small factors */
    3432           7 :   return gc_long(av, ifac_ispowerful(n));
    3433             : }
    3434             : 
    3435             : ulong
    3436       59573 : coreu_fact(GEN f)
    3437             : {
    3438       59573 :   GEN P = gel(f,1), E = gel(f,2);
    3439       59573 :   long i, l = lg(P), m = 1;
    3440      107206 :   for (i = 1; i < l; i++)
    3441             :   {
    3442       47633 :     ulong p = P[i], e = E[i];
    3443       47633 :     if (e & 1) m *= p;
    3444             :   }
    3445       59573 :   return m;
    3446             : }
    3447             : ulong
    3448       59573 : coreu(ulong n)
    3449             : {
    3450             :   pari_sp av;
    3451       59573 :   if (n == 0) return 0;
    3452       59573 :   av = avma; return gc_ulong(av, coreu_fact(factoru(n)));
    3453             : }
    3454             : GEN
    3455      706779 : core(GEN n)
    3456             : {
    3457      706779 :   pari_sp av = avma;
    3458             :   GEN m, F;
    3459             :   ulong p;
    3460             :   long i, l, v;
    3461             :   forprime_t S;
    3462             : 
    3463      706779 :   if ((F = check_arith_all(n, "core")))
    3464             :   {
    3465      646895 :     GEN p, x, P = gel(F,1), E = gel(F,2);
    3466      646895 :     long j = 1;
    3467      646895 :     if (lg(P) == 1) return gen_1;
    3468      646867 :     p = gel(P,1);
    3469      646867 :     if (!signe(p)) return gen_0;
    3470      646825 :     l = lg(P); x = cgetg(l, t_VEC);
    3471     2284334 :     for (i = 1; i < l; i++)
    3472     1637520 :       if (mpodd(gel(E,i))) gel(x,j++) = gel(P,i);
    3473      646814 :     setlg(x, j); return ZV_prod(x);
    3474             :   }
    3475       59894 :   switch(lgefint(n))
    3476             :   {
    3477          28 :     case 2: return gen_0;
    3478       59517 :     case 3:
    3479       59517 :       p = coreu(uel(n,2));
    3480       59517 :       return signe(n) > 0? utoipos(p): utoineg(p);
    3481             :   }
    3482             : 
    3483         349 :   m = signe(n) < 0? gen_m1: gen_1;
    3484         349 :   n = absi_shallow(n);
    3485         350 :   u_forprime_init(&S, 2, tridiv_bound(n));
    3486     5020465 :   while ((p = u_forprime_next_fast(&S)))
    3487             :   {
    3488             :     int stop;
    3489     5020136 :     v = Z_lvalrem_stop(&n, p, &stop);
    3490     5020136 :     if (v)
    3491             :     {
    3492         963 :       if (v & 1) m = muliu(m, p);
    3493         963 :       if (stop)
    3494             :       {
    3495          21 :         if (!is_pm1(n)) m = mulii(m, n);
    3496          21 :         return gerepileuptoint(av, m);
    3497             :       }
    3498             :     }
    3499             :   }
    3500         329 :   l = lg(primetab);
    3501         823 :   for (i = 1; i < l; i++)
    3502             :   {
    3503         502 :     GEN q = gel(primetab,i);
    3504         502 :     v = Z_pvalrem(n, q, &n);
    3505         502 :     if (v)
    3506             :     {
    3507          32 :       if (v & 1) m = mulii(m, q);
    3508          32 :       if (is_pm1(n)) return gerepileuptoint(av, m);
    3509             :     }
    3510             :   }
    3511         321 :   if (ifac_isprime(n)) { m = mulii(m, n); return gerepileuptoint(av, m); }
    3512         311 :   if (m == gen_1) n = icopy(n); /* ifac_core destroys n */
    3513             :   /* large composite without small factors */
    3514         311 :   return gerepileuptoint(av, mulii(m, ifac_core(n)));
    3515             : }
    3516             : 
    3517             : long
    3518           0 : Z_issmooth(GEN m, ulong lim)
    3519             : {
    3520           0 :   pari_sp av = avma;
    3521           0 :   ulong p = 2;
    3522             :   forprime_t S;
    3523           0 :   u_forprime_init(&S, 2, lim);
    3524           0 :   while ((p = u_forprime_next_fast(&S)))
    3525             :   {
    3526             :     int stop;
    3527           0 :     (void)Z_lvalrem_stop(&m, p, &stop);
    3528           0 :     if (stop) return gc_long(av, abscmpiu(m,lim) <= 0);
    3529             :   }
    3530           0 :   return gc_long(av,0);
    3531             : }
    3532             : 
    3533             : GEN
    3534      198068 : Z_issmooth_fact(GEN m, ulong lim)
    3535             : {
    3536      198068 :   pari_sp av = avma;
    3537             :   GEN F, P, E;
    3538             :   ulong p;
    3539      198068 :   long i = 1, l = expi(m)+1;
    3540             :   forprime_t S;
    3541      198047 :   P = cgetg(l, t_VECSMALL);
    3542      198020 :   E = cgetg(l, t_VECSMALL); F = mkmat2(P,E);
    3543      197902 :   if (l == 1) return F; /* m == 1 */
    3544      197860 :   u_forprime_init(&S, 2, lim);
    3545    50684897 :   while ((p = u_forprime_next_fast(&S)))
    3546             :   {
    3547             :     long v;
    3548             :     int stop;
    3549    50592443 :     if ((v = Z_lvalrem_stop(&m, p, &stop)))
    3550             :     {
    3551      752092 :       P[i] = p;
    3552      752092 :       E[i] = v; i++;
    3553      752092 :       if (stop)
    3554             :       {
    3555      136040 :         if (abscmpiu(m,lim) > 0) break;
    3556      115460 :         if (m[2] > 1) { P[i] = m[2]; E[i] = 1; i++; }
    3557      115460 :         setlg(P, i);
    3558      115710 :         setlg(E, i); return gc_const((pari_sp)F, F);
    3559             :       }
    3560             :     }
    3561             :   }
    3562       70200 :   return gc_NULL(av);
    3563             : }
    3564             : 
    3565             : /***********************************************************************/
    3566             : /**                                                                   **/
    3567             : /**       COMPUTING THE MATRIX OF PRIME DIVISORS AND EXPONENTS        **/
    3568             : /**                                                                   **/
    3569             : /***********************************************************************/
    3570             : static GEN
    3571      274763 : aux_end(GEN M, GEN n, long nb)
    3572             : {
    3573      274763 :   GEN P,E, z = (GEN)avma;
    3574             :   long i;
    3575             : 
    3576      274763 :   guncloneNULL(n);
    3577      274763 :   P = cgetg(nb+1,t_COL);
    3578      274763 :   E = cgetg(nb+1,t_COL);
    3579     1706548 :   for (i=nb; i; i--)
    3580             :   { /* allow a stackdummy in the middle */
    3581     1559347 :     while (typ(z) != t_INT) z += lg(z);
    3582     1431785 :     gel(E,i) = z; z += lg(z);
    3583     1431785 :     gel(P,i) = z; z += lg(z);
    3584             :   }
    3585      274763 :   gel(M,1) = P;
    3586      274763 :   gel(M,2) = E;
    3587      274763 :   return sort_factor(M, (void*)&abscmpii, cmp_nodata);
    3588             : }
    3589             : 
    3590             : static void
    3591     1426663 : STORE(long *nb, GEN x, long e) { (*nb)++; (void)x; (void)utoipos(e); }
    3592             : static void
    3593     1364862 : STOREu(long *nb, ulong x, long e) { STORE(nb, utoipos(x), e); }
    3594             : static void
    3595       61653 : STOREi(long *nb, GEN x, long e) { STORE(nb, icopy(x), e); }
    3596             : /* no prime less than p divides n; return 1 if factored completely */
    3597             : static int
    3598       39264 : special_primes(GEN n, ulong p, long *nb, GEN T)
    3599             : {
    3600       39264 :   long i, l = lg(T);
    3601       39264 :   if (l > 1)
    3602             :   { /* pp = square of biggest p tried so far */
    3603         540 :     long pp[] = { evaltyp(t_INT)|_evallg(4), 0,0,0 };
    3604         540 :     pari_sp av = avma; affii(sqru(p), pp); set_avma(av);
    3605             : 
    3606        1184 :     for (i = 1; i < l; i++)
    3607         777 :       if (dvdiiz(n, gel(T,i), n))
    3608             :       {
    3609         329 :         long k = 1; while (dvdiiz(n, gel(T,i), n)) k++;
    3610         231 :         STOREi(nb, gel(T,i), k);
    3611         231 :         if (abscmpii(pp, n) > 0)
    3612             :         {
    3613         133 :           if (!is_pm1(n)) STOREi(nb, n, 1);
    3614         133 :           return 1;
    3615             :         }
    3616             :       }
    3617             :   }
    3618       39131 :   return 0;
    3619             : }
    3620             : 
    3621             : /* factor(sn*|n|), where sn = -1 or 1.
    3622             :  * all != 0 : only look for prime divisors < all. If pU is not NULL,
    3623             :  * set it to unfactored composite */
    3624             : static GEN
    3625    14476560 : ifactor_sign(GEN n, ulong all, long hint, long sn, GEN *pU)
    3626             : {
    3627             :   GEN M, N;
    3628             :   pari_sp av;
    3629    14476560 :   long nb = 0, nb0 = -1, i;
    3630             :   ulong lim;
    3631             :   forprime_t T;
    3632             : 
    3633    14476560 :   if (lgefint(n) == 3)
    3634             :   { /* small integer */
    3635    14201873 :     GEN f, Pf, Ef, P, E, F = cgetg(3, t_MAT);
    3636             :     ulong U1, U2;
    3637             :     long l;
    3638    14202039 :     av = avma;
    3639             :     /* enough room to store <= 15 primes and exponents (OK if n < 2^64) */
    3640    14202039 :     (void)new_chunk((15*3 + 15 + 1) * 2);
    3641    14202008 :     f = factoru_sign(uel(n,2), all, hint, pU? &U1: NULL, pU? &U2: NULL);
    3642    14202290 :     set_avma(av);
    3643    14202297 :     Pf = gel(f,1);
    3644    14202297 :     Ef = gel(f,2);
    3645    14202297 :     l = lg(Pf);
    3646    14202297 :     if (sn < 0)
    3647             :     { /* add sign */
    3648        6523 :       long L = l+1;
    3649        6523 :       gel(F,1) = P = cgetg(L, t_COL);
    3650        6523 :       gel(F,2) = E = cgetg(L, t_COL);
    3651        6523 :       gel(P,1) = gen_m1; P++;
    3652        6523 :       gel(E,1) = gen_1;  E++;
    3653             :     }
    3654             :     else
    3655             :     {
    3656    14195774 :       gel(F,1) = P = cgetg(l, t_COL);
    3657    14195750 :       gel(F,2) = E = cgetg(l, t_COL);
    3658             :     }
    3659    44353771 :     for (i = 1; i < l; i++)
    3660             :     {
    3661    30151512 :       gel(P,i) = utoipos(Pf[i]);
    3662    30151489 :       gel(E,i) = utoipos(Ef[i]);
    3663             :     }
    3664    14202259 :     if (pU) *pU = U1 == 1? NULL: mkvec2(utoipos(U1), utoipos(U2));
    3665    14202250 :     return F;
    3666             :   }
    3667      274687 :   if (pU) *pU = NULL;
    3668      274687 :   M = cgetg(3,t_MAT);
    3669      274763 :   if (sn < 0) STORE(&nb, utoineg(1), 1);
    3670      274763 :   if (is_pm1(n)) return aux_end(M,NULL,nb);
    3671             : 
    3672      274763 :   n = N = gclone(n); setabssign(n);
    3673             :   /* trial division bound; look for primes <= lim; nb is the number of
    3674             :    * distinct prime factors so far; if nb0 >= 0, it records the value of nb
    3675             :    * for which we made a successful compositeness test: if later nb = nb0,
    3676             :    * we know that n is composite */
    3677      274763 :   lim = 1;
    3678      274763 :   if (!all || all > 2)
    3679             :   { /* trial divide p < all if all != 0, else p <= tridiv_bound() >= 2^14 */
    3680             :     ulong maxp, p;
    3681             :     pari_sp av2;
    3682      274749 :     i = vali(n);
    3683      274749 :     if (i)
    3684             :     {
    3685      128995 :       STOREu(&nb, 2, i);
    3686      128995 :       av = avma; affii(shifti(n,-i), n); set_avma(av);
    3687             :     }
    3688      274749 :     if (is_pm1(n)) return aux_end(M,n,nb);
    3689      274614 :     lim = all? all-1: tridiv_bound(n);
    3690      274614 :     maxp = maxprime();
    3691      274614 :     if (!(hint & 16) && lim >= 128) /* expu(lim) >= 7 */
    3692       38094 :     { /* fast trial division */
    3693      243102 :       GEN nr, PR = prodprimes();
    3694      243102 :       long nPR = lg(PR)-1, b = minss(nPR, expu(lim)-6);
    3695      243102 :       av = avma; nr = gcdii(gel(PR,b), n);
    3696      243102 :       if (is_pm1(nr)) { set_avma(av); av2 = av; }
    3697             :       else
    3698             :       {
    3699      240357 :         GEN F = ifactor_sign(nr, all, 1 + 2 + 16, 1, NULL), P = gel(F,1);
    3700      240357 :         long l = lg(P);
    3701      240357 :         av2 = avma;
    3702     1186493 :         for (i = 1; i < l; i++)
    3703             :         {
    3704      947073 :           pari_sp av3 = avma;
    3705      947073 :           GEN gp = gel(P,i);
    3706      947073 :           ulong p = gp[2];
    3707             :           long k;
    3708      947073 :           if (lgefint(gp)>3 || (all && p>=all)) break; /* may occur for last p */
    3709      946136 :           k = Z_lvalrem(n, p, &n); /* > 0 */
    3710      946136 :           affii(n, N); n = N; set_avma(av3);
    3711      946136 :           STOREu(&nb, p, k);
    3712             :         }
    3713      240357 :         if (is_pm1(n))
    3714             :         {
    3715      205008 :           stackdummy(av, av2);
    3716      205008 :           return aux_end(M,n,nb);
    3717             :         }
    3718             :       }
    3719             :     }
    3720             :     else
    3721             :     { /* naive trial division */
    3722       31512 :       av = avma;
    3723       31512 :       u_forprime_init(&T, 3, minss(lim, maxp)); av2 = avma;
    3724             :       /* first pass: known to fit in private prime table */
    3725    38461630 :       while ((p = u_forprime_next_fast(&T)))
    3726             :       {
    3727    38460474 :         pari_sp av3 = avma;
    3728             :         int stop;
    3729    38460474 :         long k = Z_lvalrem_stop(&n, p, &stop);
    3730    38460474 :         if (k)
    3731             :         {
    3732      289731 :           affii(n, N); n = N; set_avma(av3);
    3733      289731 :           STOREu(&nb, p, k);
    3734             :         }
    3735             :         /* prodeuler(p=2,16381,1-1/p) ~ 0.0578; if probability of being prime
    3736             :          * knowing P^-(n) > 16381 is at least 10%, try BPSW */
    3737    38460474 :         if (!stop && p == 16381)
    3738             :         {
    3739        3296 :           if (bit_accuracy_mul(lgefint(n), 0.0578 * M_LN2) < 10)
    3740        3296 :           { nb0 = nb; stop = ifac_isprime(n); }
    3741             :         }
    3742    38460474 :         if (stop)
    3743             :         {
    3744       30356 :           if (!is_pm1(n)) STOREi(&nb, n, 1);
    3745       30356 :           stackdummy(av, av2);
    3746       30356 :           return aux_end(M,n,nb);
    3747             :         }
    3748             :       }
    3749             :     }
    3750       39250 :     stackdummy(av, av2);
    3751       39250 :     if (lim > maxp)
    3752             :     { /* second pass, usually empty: outside private prime table */
    3753        4677 :       av = avma; u_forprime_init(&T, maxp+1, lim); av2 = avma;
    3754        4677 :       while ((p = u_forprime_next(&T)))
    3755             :       {
    3756           0 :         pari_sp av3 = avma;
    3757             :         int stop;
    3758           0 :         long k = Z_lvalrem_stop(&n, p, &stop);
    3759           0 :         if (k)
    3760             :         {
    3761           0 :           affii(n, N); n = N; set_avma(av3);
    3762           0 :           STOREu(&nb, p, k);
    3763             :         }
    3764           0 :         if (stop)
    3765             :         {
    3766           0 :           if (!is_pm1(n)) STOREi(&nb, n, 1);
    3767           0 :           stackdummy(av, av2);
    3768           0 :           return aux_end(M,n,nb);
    3769             :         }
    3770             :       }
    3771        4677 :       stackdummy(av, av2);
    3772             :     }
    3773             :   }
    3774       39264 :   if (special_primes(n, lim, &nb, primetab)) return aux_end(M,n, nb);
    3775             :   /* if nb > nb0 (includes nb0 = -1) we don't know that n is composite */
    3776       39131 :   if (all)
    3777             :   { /* smallfact: look for easy pure powers then stop. Cf Z_isanypower */
    3778             :     GEN x;
    3779             :     long k;
    3780       31071 :     av = avma;
    3781       31071 :     k = isanypower_nosmalldiv(n, &x); /* may miss a power if all < 2^14 */
    3782       31071 :     if (k > 1) { affii(x, n); nb0 = -1; }
    3783       31071 :     if (pU)
    3784             :     {
    3785             :       GEN F;
    3786       12766 :       if (abscmpiu(n, lim) <= 0
    3787       12766 :           || cmpii(n, sqru(lim)) <= 0
    3788        8040 :           || (all >= (1<<14)
    3789        7265 :               && (nb>nb0 && bit_accuracy(lgefint(n))<2048 && ifac_isprime(n))))
    3790       12766 :       { set_avma(av); STOREi(&nb, n, k); return aux_end(M,n, nb); }
    3791        5282 :       set_avma(av); F = aux_end(M, NULL, nb); /* don't destroy n */
    3792        5282 :       *pU = mkvec2(icopy(n), utoipos(k)); /* composite cofactor */
    3793        5282 :       gunclone(n); return F;
    3794             :     }
    3795       18305 :     set_avma(av); STOREi(&nb, n, k);
    3796       18305 :     if (DEBUGLEVEL >= 2) {
    3797           0 :       pari_warn(warner,
    3798           0 :         "IFAC: untested %ld-bit integer declared prime", expi(n)+1);
    3799           0 :       if (expi(n) <= 256) err_printf("\t%Ps\n", n);
    3800             :     }
    3801             :   }
    3802        8060 :   else if (nb > nb0 && ifac_isprime(n)) STOREi(&nb, n, 1);
    3803        2643 :   else nb += ifac_decomp(n, hint);
    3804       26365 :   return aux_end(M,n, nb);
    3805             : }
    3806             : 
    3807             : static GEN
    3808     9551397 : ifactor(GEN n, ulong all, long hint)
    3809             : {
    3810     9551397 :   long s = signe(n);
    3811     9551397 :   if (!s) retmkmat2(mkcol(gen_0), mkcol(gen_1));
    3812     9551348 :   return ifactor_sign(n, all, hint, s, NULL);
    3813             : }
    3814             : 
    3815             : int
    3816        7194 : ifac_next(GEN *part, GEN *p, long *e)
    3817             : {
    3818        7194 :   GEN here = ifac_main(part);
    3819        7194 :   if (here == gen_0) { *p = NULL; *e = 1; return 0; }
    3820        7180 :   if (!here) { *p = NULL; *e = 0; return 0; }
    3821        4832 :   *p = VALUE(here);
    3822        4832 :   *e = EXPON(here)[2];
    3823        4832 :   ifac_delete(here); return 1;
    3824             : }
    3825             : 
    3826             : /* see before ifac_crack for current semantics of 'hint' (factorint's 'flag') */
    3827             : GEN
    3828       10290 : factorint(GEN n, long flag)
    3829             : {
    3830             :   GEN F;
    3831       10290 :   if ((F = check_arith_all(n,"factorint"))) return gcopy(F);
    3832       10276 :   return ifactor(n,0,flag);
    3833             : }
    3834             : 
    3835             : GEN
    3836       20364 : Z_factor_limit(GEN n, ulong all)
    3837             : {
    3838       20364 :   if (!all) all = GP_DATA->primelimit + 1;
    3839       20364 :   return ifactor(n, all, decomp_default_hint);
    3840             : }
    3841             : GEN
    3842      888360 : absZ_factor_limit_strict(GEN n, ulong all, GEN *pU)
    3843             : {
    3844             :   GEN F, U;
    3845      888360 :   if (!signe(n))
    3846             :   {
    3847           0 :     if (pU) *pU = NULL;
    3848           0 :     retmkmat2(mkcol(gen_0), mkcol(gen_1));
    3849             :   }
    3850      888360 :   if (!all) all = GP_DATA->primelimit + 1;
    3851      888360 :   F = ifactor_sign(n, all, decomp_default_hint, 1, &U);
    3852      888399 :   if (pU) *pU = U;
    3853      888399 :   return F;
    3854             : }
    3855             : GEN
    3856      290440 : absZ_factor_limit(GEN n, ulong all)
    3857             : {
    3858      290440 :   if (!signe(n)) retmkmat2(mkcol(gen_0), mkcol(gen_1));
    3859      290440 :   if (!all) all = GP_DATA->primelimit + 1;
    3860      290440 :   return ifactor_sign(n, all, decomp_default_hint, 1, NULL);
    3861             : }
    3862             : GEN
    3863     9520709 : Z_factor(GEN n)
    3864     9520709 : { return ifactor(n,0,decomp_default_hint); }
    3865             : GEN
    3866     3505813 : absZ_factor(GEN n)
    3867             : {
    3868     3505813 :   if (!signe(n)) retmkmat2(mkcol(gen_0), mkcol(gen_1));
    3869     3505799 :   return ifactor_sign(n, 0, decomp_default_hint, 1, NULL);
    3870             : }
    3871             : /* Factor until the unfactored part is smaller than limit. Return the
    3872             :  * factored part. Hence factorback(output) may be smaller than n */
    3873             : GEN
    3874          98 : Z_factor_until(GEN n, GEN limit)
    3875             : {
    3876          98 :   pari_sp av = avma;
    3877          98 :   long s = signe(n), eq;
    3878             :   GEN q, F, U;
    3879             : 
    3880          98 :   if (!s) retmkmat2(mkcol(gen_0), mkcol(gen_1));
    3881          98 :   F = ifactor_sign(n, tridiv_bound(n), decomp_default_hint, s, &U);
    3882          98 :   if (!U) return F;
    3883          28 :   q = gel(U,1); /* composite, q^eq = unfactored part */
    3884          28 :   eq = itou(gel(U,2));
    3885          28 :   if (cmpii(eq == 1? q: powiu(q, eq), limit) > 0)
    3886             :   { /* factor further */
    3887          21 :     long l2 = expi(q)+1;
    3888             :     GEN P2, E2, F2, part;
    3889          21 :     if (eq > 1) limit = sqrtnint(limit, eq);
    3890          21 :     P2 = coltrunc_init(l2);
    3891          21 :     E2 = coltrunc_init(l2); F2 = mkmat2(P2,E2);
    3892          21 :     part = ifac_start(icopy(q), 0); /* ifac_next would destroy q */
    3893             :     for(;;)
    3894           0 :     {
    3895             :       long e;
    3896             :       GEN p;
    3897          21 :       if (!ifac_next(&part,&p,&e)) break;
    3898          21 :       vectrunc_append(P2, p);
    3899          21 :       vectrunc_append(E2, utoipos(e * eq));
    3900          21 :       q = diviiexact(q, powiu(p, e));
    3901          21 :       if (cmpii(q, limit) <= 0) break;
    3902             :     }
    3903          21 :     F2 = sort_factor(F2, (void*)&abscmpii, cmp_nodata);
    3904          21 :     F = merge_factor(F, F2, (void*)&abscmpii, cmp_nodata);
    3905             :   }
    3906          28 :   return gerepilecopy(av, F);
    3907             : }
    3908             : 
    3909             : static void
    3910   110643360 : matsmalltrunc_append(GEN m, ulong p, ulong e)
    3911             : {
    3912   110643360 :   GEN P = gel(m,1), E = gel(m,2);
    3913   110643360 :   long l = lg(P);
    3914   110643360 :   P[l] = p; lg_increase(P);
    3915   110668537 :   E[l] = e; lg_increase(E);
    3916   110644846 : }
    3917             : static GEN
    3918    44781248 : matsmalltrunc_init(long l)
    3919             : {
    3920    44781248 :   GEN P = vecsmalltrunc_init(l);
    3921    44526237 :   GEN E = vecsmalltrunc_init(l); return mkvec2(P,E);
    3922             : }
    3923             : 
    3924             : /* return optimal N s.t. omega(b) <= N for all b <= x */
    3925             : long
    3926       59580 : maxomegau(ulong x)
    3927             : { /* P=primes(15); for(i=1,15, print([i, vecprod(P[1..i])])) */
    3928       59580 :   if (x < 30030UL)/* rare trivial cases */
    3929             :   {
    3930       33771 :     if (x < 2UL) return 0;
    3931       15516 :     if (x < 6UL) return 1;
    3932        9916 :     if (x < 30UL) return 2;
    3933        9209 :     if (x < 210UL) return 3;
    3934        8971 :     if (x < 2310UL) return 4;
    3935        8173 :     return 5;
    3936             :   }
    3937       25809 :   if (x < 510510UL) return 6; /* most frequent case */
    3938       13776 :   if (x < 9699690UL) return 7;
    3939           7 :   if (x < 223092870UL) return 8;
    3940             : #ifdef LONG_IS_64BIT
    3941           6 :   if (x < 6469693230UL) return 9;
    3942           0 :   if (x < 200560490130UL) return 10;
    3943           0 :   if (x < 7420738134810UL) return 11;
    3944           0 :   if (x < 304250263527210UL) return 12;
    3945           0 :   if (x < 13082761331670030UL) return 13;
    3946           0 :   if (x < 614889782588491410UL) return 14;
    3947           0 :   return 15;
    3948             : #else
    3949           1 :   return 9;
    3950             : #endif
    3951             : }
    3952             : /* return optimal N s.t. omega(b) <= N for all odd b <= x */
    3953             : long
    3954        3027 : maxomegaoddu(ulong x)
    3955             : { /* P=primes(15+1); for(i=1,15, print([i, vecprod(P[2..i+1])])) */
    3956        3027 :   if (x < 255255UL)/* rare trivial cases */
    3957             :   {
    3958        1815 :     if (x < 3UL) return 0;
    3959        1815 :     if (x < 15UL) return 1;
    3960        1815 :     if (x < 105UL) return 2;
    3961        1815 :     if (x < 1155UL) return 3;
    3962        1787 :     if (x < 15015UL) return 4;
    3963        1787 :     return 5;
    3964             :   }
    3965        1212 :   if (x < 4849845UL) return 6; /* most frequent case */
    3966           0 :   if (x < 111546435UL) return 7;
    3967           0 :   if (x < 3234846615UL) return 8;
    3968             : #ifdef LONG_IS_64BIT
    3969           0 :   if (x < 100280245065UL) return 9;
    3970           0 :   if (x < 3710369067405UL) return 10;
    3971           0 :   if (x < 152125131763605UL) return 11;
    3972           0 :   if (x < 6541380665835015UL) return 12;
    3973           0 :   if (x < 307444891294245705UL) return 13;
    3974           0 :   if (x < 16294579238595022365UL) return 14;
    3975           0 :   return 15;
    3976             : #else
    3977           0 :   return 9;
    3978             : #endif
    3979             : }
    3980             : 
    3981             : /* If a <= c <= b , factoru(c) = L[c-a+1] */
    3982             : GEN
    3983       31957 : vecfactoru_i(ulong a, ulong b)
    3984             : {
    3985       31957 :   ulong k, p, n = b-a+1, N = maxomegau(b) + 1;
    3986       31957 :   GEN v = const_vecsmall(n, 1);
    3987       31958 :   GEN L = cgetg(n+1, t_VEC);
    3988             :   forprime_t T;
    3989    21260261 :   for (k = 1; k <= n; k++) gel(L,k) = matsmalltrunc_init(N);
    3990       31957 :   u_forprime_init(&T, 2, usqrt(b));
    3991      876674 :   while ((p = u_forprime_next(&T)))
    3992             :   { /* p <= sqrt(b) */
    3993      855053 :     ulong pk = p, K = ulogint(b, p);
    3994     2966042 :     for (k = 1; k <= K; k++)
    3995             :     {
    3996     2121325 :       ulong j, t = a / pk, ap = t * pk;
    3997     2121325 :       if (ap < a) { ap += pk; t++; }
    3998             :       /* t = (j+a-1) \ pk */
    3999     2121325 :       t %= p;
    4000    60530663 :       for (j = ap-a+1; j <= n; j += pk)
    4001             :       {
    4002    58419691 :         if (t) { v[j] *= pk; matsmalltrunc_append(gel(L,j), p,k); }
    4003    58409338 :         if (++t == p) t = 0;
    4004             :       }
    4005     2110972 :       pk *= p;
    4006             :     }
    4007             :   }
    4008             :   /* complete factorisation of non-sqrt(b)-smooth numbers */
    4009    21492741 :   for (k = 1, N = a; k <= n; k++, N++)
    4010    21468640 :     if (uel(v,k) != N) matsmalltrunc_append(gel(L,k), N/uel(v,k),1UL);
    4011       27311 :   return L;
    4012             : }
    4013             : GEN
    4014           0 : vecfactoru(ulong a, ulong b)
    4015             : {
    4016           0 :   pari_sp av = avma;
    4017           0 :   return gerepilecopy(av, vecfactoru_i(a,b));
    4018             : }
    4019             : 
    4020             : /* Assume a and b odd, return L s.t. L[k] = factoru(a + 2*(k-1))
    4021             :  * If a <= c <= b odd, factoru(c) = L[(c-a)>>1 + 1] */
    4022             : GEN
    4023        3027 : vecfactoroddu_i(ulong a, ulong b)
    4024             : {
    4025        3027 :   ulong k, p, n = ((b-a)>>1) + 1, N = maxomegaoddu(b) + 1;
    4026        3027 :   GEN v = const_vecsmall(n, 1);
    4027        3027 :   GEN L = cgetg(n+1, t_VEC);
    4028             :   forprime_t T;
    4029             : 
    4030    24067783 :   for (k = 1; k <= n; k++) gel(L,k) = matsmalltrunc_init(N);
    4031        3027 :   u_forprime_init(&T, 3, usqrt(b));
    4032      239248 :   while ((p = u_forprime_next(&T)))
    4033             :   { /* p <= sqrt(b) */
    4034      253447 :     ulong pk = p, K = ulogint(b, p);
    4035      845890 :     for (k = 1; k <= K; k++)
    4036             :     {
    4037      609669 :       ulong j, t = (a / pk) | 1UL, ap = t * pk;
    4038             :       /* t and ap are odd, ap multiple of pk = p^k */
    4039      609669 :       if (ap < a) { ap += pk<<1; t+=2; }
    4040             :       /* c=t*p^k by steps of 2*p^k; factorization of c*=p^k if (t,p)=1 */
    4041      609669 :       t %= p;
    4042    44349816 :       for (j = ((ap-a)>>1)+1; j <= n; j += pk)
    4043             :       {
    4044    43757428 :         if (t) { v[j] *= pk; matsmalltrunc_append(gel(L,j), p,k); }
    4045    43740147 :         t += 2; if (t >= p) t -= p;
    4046             :       }
    4047      592388 :       pk *= p;
    4048             :     }
    4049             :   }
    4050             :   /* complete factorisation of non-sqrt(b)-smooth numbers */
    4051    24180620 :   for (k = 1, N = a; k <= n; k++, N+=2)
    4052    24178517 :     if (uel(v,k) != N) matsmalltrunc_append(gel(L,k), N/uel(v,k),1UL);
    4053        2103 :   return L;
    4054             : }
    4055             : GEN
    4056           0 : vecfactoroddu(ulong a, ulong b)
    4057             : {
    4058           0 :   pari_sp av = avma;
    4059           0 :   return gerepilecopy(av, vecfactoroddu_i(a,b));
    4060             : }
    4061             : 
    4062             : /* If 0 <= a <= c <= b; L[c-a+1] = factoru(c)[,1] if c squarefree, else NULL */
    4063             : GEN
    4064        7014 : vecfactorsquarefreeu(ulong a, ulong b)
    4065             : {
    4066        7014 :   ulong k, p, n = b-a+1, N = maxomegau(b) + 1;
    4067        7014 :   GEN v = const_vecsmall(n, 1);
    4068        7014 :   GEN L = cgetg(n+1, t_VEC);
    4069             :   forprime_t T;
    4070    14007238 :   for (k = 1; k <= n; k++) gel(L,k) = vecsmalltrunc_init(N);
    4071        7014 :   u_forprime_init(&T, 2, usqrt(b));
    4072      838334 :   while ((p = u_forprime_next(&T)))
    4073             :   { /* p <= sqrt(b), kill nonsquarefree */
    4074      831320 :     ulong j, pk = p*p, t = a / pk, ap = t * pk;
    4075      831320 :     if (ap < a) ap += pk;
    4076     7160090 :     for (j = ap-a+1; j <= n; j += pk) gel(L,j) = NULL;
    4077             : 
    4078      831320 :     t = a / p; ap = t * p;
    4079      831320 :     if (ap < a) { ap += p; t++; }
    4080    30551556 :     for (j = ap-a+1; j <= n; j += p, t++)
    4081    29720236 :       if (gel(L,j)) { v[j] *= p; vecsmalltrunc_append(gel(L,j), p); }
    4082             :   }
    4083             :   /* complete factorisation of non-sqrt(b)-smooth numbers */
    4084    14007238 :   for (k = 1, N = a; k <= n; k++, N++)
    4085    14000224 :     if (gel(L,k) && uel(v,k) != N) vecsmalltrunc_append(gel(L,k), N/uel(v,k));
    4086        7014 :   return L;
    4087             : }
    4088             : /* If 0 <= a <= c <= b; L[c-a+1] = factoru(c)[,1] if c squarefree and coprime
    4089             :  * to all the primes in sorted zv P, else NULL */
    4090             : GEN
    4091       20531 : vecfactorsquarefreeu_coprime(ulong a, ulong b, GEN P)
    4092             : {
    4093       20531 :   ulong k, p, n = b-a+1, sqb = usqrt(b), N = maxomegau(b) + 1;
    4094       20532 :   GEN v = const_vecsmall(n, 1);
    4095       20532 :   GEN L = cgetg(n+1, t_VEC);
    4096             :   forprime_t T;
    4097    65073428 :   for (k = 1; k <= n; k++) gel(L,k) = vecsmalltrunc_init(N);
    4098       20531 :   u_forprime_init(&T, 2, sqb);
    4099     2172076 :   while ((p = u_forprime_next(&T)))
    4100             :   { /* p <= sqrt(b), kill nonsquarefree */
    4101     2153722 :     ulong j, t, ap, bad = zv_search(P, p), pk = bad ? p: p * p;
    4102     2156063 :     t = a / pk; ap = t * pk; if (ap < a) ap += pk;
    4103    55959268 :     for (j = ap-a+1; j <= n; j += pk) gel(L,j) = NULL;
    4104     2156063 :     if (bad) continue;
    4105             : 
    4106     2117960 :     t = a / p; ap = t * p;
    4107     2117960 :     if (ap < a) { ap += p; t++; }
    4108    83676972 :     for (j = ap-a+1; j <= n; j += p, t++)
    4109    81563530 :       if (gel(L,j)) { v[j] *= p; vecsmalltrunc_append(gel(L,j), p); }
    4110             :   }
    4111       20532 :   if (uel(P,lg(P)-1) <= sqb) P = NULL;
    4112             :   /* complete factorisation of non-sqrt(b)-smooth numbers */
    4113    65343536 :   for (k = 1, N = a; k <= n; k++, N++)
    4114    65323161 :     if (gel(L,k) && uel(v,k) != N)
    4115             :     {
    4116    18929831 :       ulong q = N / uel(v,k);
    4117    18929831 :       if (!P || !zv_search(P, q)) vecsmalltrunc_append(gel(L,k), q);
    4118             :     }
    4119       20375 :   return L;
    4120             : }
    4121             : 
    4122             : GEN
    4123          49 : vecsquarefreeu(ulong a, ulong b)
    4124             : {
    4125          49 :   ulong j, k, p, n = b-a+1;
    4126          49 :   GEN L = const_vecsmall(n, 1);
    4127             :   forprime_t T;
    4128          49 :   u_forprime_init(&T, 2, usqrt(b));
    4129         462 :   while ((p = u_forprime_next(&T)))
    4130             :   { /* p <= sqrt(b), kill nonsquarefree */
    4131         413 :     ulong pk = p*p, t = a / pk, ap = t * pk;
    4132         413 :     if (ap < a) { ap += pk; t++; }
    4133             :     /* t = (j+a-1) \ pk */
    4134       21777 :     for (j = ap-a+1; j <= n; j += pk, t++) L[j] = 0;
    4135             :   }
    4136       48258 :   for (k = j = 1; k <= n; k++)
    4137       48209 :     if (L[k]) L[j++] = a+k-1;
    4138          49 :   setlg(L,j); return L;
    4139             : }

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