These functions are by definition functions whose natural domain of definition is either ℤ (or ℤ > 0). The way these functions are used is completely different from transcendental functions in that there are no automatic type conversions: in general only integers are accepted as arguments. An integer argument N can be given in the following alternate formats:
* t_MAT: its factorization fa = factor(N),
* t_VEC: a pair [N, fa] giving both the integer and
its factorization.
This allows to compute different arithmetic functions at a given N while factoring the latter only once.
? N = 10!; faN = factor(N);
? eulerphi(N)
%2 = 829440
? eulerphi(faN)
%3 = 829440
? eulerphi(S = [N, faN])
%4 = 829440
? sigma(S)
%5 = 15334088
All arithmetic functions in the narrow sense of the word — Euler's
totient function, the Moebius function,
the sums over divisors or powers of divisors etc. — call, after trial
division by small primes, the same versatile factoring machinery described
under factorint. It includes Shanks SQUFOF, Pollard Rho,
ECM and MPQS stages, and has an early exit option for the
functions moebius and (the integer function underlying)
issquarefree. This machinery relies on a fairly strong
probabilistic primality test, see ispseudoprime, but you may also set
default(factor_proven, 1)
to ensure that all tentative factorizations are fully proven. This should not slow down PARI too much, unless prime numbers with hundreds of decimal digits occur frequently in your application.
The following functions compute the order of an element in a finite group:
ellorder (the rational points on an elliptic curve defined over a
finite field), fforder (the multiplicative group of a finite field),
znorder (the invertible elements in ℤ/nℤ). The following functions
compute discrete logarithms in the same groups (whenever this is meaningful)
elllog, fflog, znlog.
All such functions allow an optional argument specifying an integer N, representing the order of the group. (The order functions also allows any nonzero multiple of the order, with a minor loss of efficiency.) That optional argument follows the same format as given above:
* t_INT: the integer N,
* t_MAT: the factorization fa = factor(N),
* t_VEC: this is the preferred format and provides both the
integer N and its factorization in a two-component vector
[N, fa].
When the group is fixed and many orders or discrete logarithms will be computed, it is much more efficient to initialize this data once and pass it to the relevant functions, as in
? p = nextprime(10^30); ? v = [p-1, factor(p-1)]; \\ data for discrete log & order computations ? znorder(Mod(2,p), v) %3 = 500000000000000000000000000028 ? g = znprimroot(p); ? znlog(2, g, v) %5 = 543038070904014908801878611374
The finite abelian group G = (ℤ/Nℤ)* can be written G = ⨁ i ≤ n (ℤ/diℤ) gi, with dn | ... | d2 | d1 (SNF condition), all di > 0, and ∏i di = φ(N).
The positivity and SNF condition make the di unique, but the generators gi, of respective order di, are definitely not unique. The ⨁ notation means that all elements of G can be written uniquely as ∏i gini where ni ∈ ℤ/diℤ. The gi are the so-called SNF generators of G.
* a character on the abelian group ⨁ j (ℤ/djℤ) gj is given by a row vector χ = [a1,...,an] of integers 0 ≤ ai < di such that χ(gj) = e(aj / dj) for all j, with the standard notation e(x) := exp(2iπ x). In other words, χ(∏j gjnj) = e(∑j aj nj / dj).
This will be generalized to more general abelian groups in later sections (Hecke characters), but in the present case of (ℤ/Nℤ)*, there is a useful alternate convention : namely, it is not necessary to impose the SNF condition and we can use Chinese remainders instead. If N = ∏ pep is the factorization of N into primes, the so-called Conrey generators of G are the generators of the (ℤ/pepℤ)* lifted to (ℤ/Nℤ)* by requesting that they be congruent to 1 modulo N/pep (for p odd we take the smallest positive primitive root mod p2, and for p = 2 we take -1 if e2 > 1 and additionally 5 if e2 > 2). We can again write G = ⨁ i ≤ n (ℤ/Diℤ) Gi, where again ∏i Di = φ(N). These generators don't satisfy the SNF condition in general since their orders are now (p-1)pep-1 for p odd; for p = 2, the generator -1 has order 2 and 5 has order 2e2-2 (e2 > 2). Nevertheless, any m ∈ (ℤ/Nℤ)* can be uniquely decomposed as m = ∏j Gimi for some mi modulo Di and we can define a character by χ(Gj) = e(mj / Dj) for all j.
* The column vector of the mj, 0 ≤ mj < Dj is
called the Conrey logarithm of m (discrete logarithm in terms of the
Conrey generators). Note that discrete logarithms in PARI/GP are always
expressed as t_COLs.
* The attached character is called the Conrey character attached to m.
To sum up a Dirichlet character can be defined by a t_INTMOD
Mod(m, N), a t_INT lift (the Conrey label m),
a t_COL (the Conrey logarithm of m, in terms of the Conrey
generators) or a t_VEC (in terms of the SNF generators). The t_COL
format, i.e. Conrey logarithms, is the preferred (fastest) representation.
Concretely, this works as follows:
G = znstar(N, 1) initializes (ℤ/Nℤ)*, which must be given as
first arguments to all functions handling Dirichlet characters.
znconreychar transforms t_INT, t_INTMOD and t_COL to a SNF
character.
znconreylog transforms t_INT, t_INTMOD and t_VEC
to a Conrey logarithm.
znconreyexp transforms t_VEC and t_COL to a Conrey label.
Also available are charconj, chardiv, charmul,
charker, chareval, charorder, zncharinduce,
znconreyconductor (also computes the primitive character attached to
the input character). The prefix char indicates that the function
applies to all characters, the prefix znchar that it is specific to
Dirichlet characters (on (ℤ/Nℤ)*) and the prefix znconrey that it
is specific to Conrey representation.
Adds the integers contained in the
vector x (or the single integer x) to a special table of
"user-defined primes", and returns that table. Whenever factor is
subsequently called, it will trial divide by the elements in this table.
If x is empty or omitted, just returns the current list of extra
primes.
? addprimes(37975227936943673922808872755445627854565536638199)
? factor(15226050279225333605356183781326374297180681149613806\
88657908494580122963258952897654000350692006139)
%2 =
[37975227936943673922808872755445627854565536638199 1]
[40094690950920881030683735292761468389214899724061 1]
? ##
*** last result computed in 0 ms.
The entries in x must be primes: there is no internal check, even if
the factor_proven default is set. To remove primes from the list use
removeprimes.
The library syntax is GEN addprimes(GEN x = NULL).
Using variants of the extended Euclidean algorithm, returns a rational approximation a/b to x, whose denominator is limited by B, if present. If B is omitted, returns the best approximation affordable given the input accuracy; if you are looking for true rational numbers, presumably approximated to sufficient accuracy, you should first try that option. Otherwise, B must be a positive real scalar (impose 0 < b ≤ B).
* If x is a t_REAL or a t_FRAC, this function uses continued
fractions.
? bestappr(Pi, 100) %1 = 22/7 ? bestappr(0.1428571428571428571428571429) %2 = 1/7 ? bestappr([Pi, sqrt(2) + 'x], 10^3) %3 = [355/113, x + 1393/985]
By definition, a/b is the best rational approximation to x if |b x - a| < |v x - u| for all integers (u,v) with 0 < v ≤ B. (Which implies that n/d is a convergent of the continued fraction of x.)
* If x is a t_INTMOD modulo N or a t_PADIC of precision N =
pk, this function performs rational modular reconstruction modulo N. The
routine then returns the unique rational number a/b in coprime integers
|a| < N/2B and b ≤ B which is congruent to x modulo N. Omitting
B amounts to choosing it of the order of sqrt{N/2}. If rational
reconstruction is not possible (no suitable a/b exists), returns [].
? bestappr(Mod(18526731858, 11^10)) %1 = 1/7 ? bestappr(Mod(18526731858, 11^20)) %2 = [] ? bestappr(3 + 5 + 3*5^2 + 5^3 + 3*5^4 + 5^5 + 3*5^6 + O(5^7)) %2 = -1/3
In most concrete uses, B is a prime power and we performed Hensel lifting to obtain x.
The function applies recursively to components of complex objects (polynomials, vectors,...). If rational reconstruction fails for even a single entry, returns [].
The library syntax is GEN bestappr(GEN x, GEN B = NULL).
Using variants of the extended Euclidean algorithm (Padé approximants), returns a rational function approximation a/b to x, whose denominator is limited by B, if present. If B is omitted, return the best approximation affordable given the input accuracy; if you are looking for true rational functions, presumably approximated to sufficient accuracy, you should first try that option. Otherwise, B must be a nonnegative real (impose 0 ≤ degree(b) ≤ B).
* If x is a t_POLMOD modulo N this function performs rational
modular reconstruction modulo N. The routine then returns the unique
rational function a/b in coprime polynomials, with degree(b) ≤ B
and degree(a) minimal, which is congruent to x modulo N.
Omitting B amounts to choosing it equal to the floor of
degree(N) / 2. If rational reconstruction is not possible (no
suitable a/b exists), returns [].
? T = Mod(x^3 + x^2 + x + 3, x^4 - 2); ? bestapprPade(T) %2 = (2*x - 1)/(x - 1) ? U = Mod(1 + x + x^2 + x^3 + x^5, x^9); ? bestapprPade(U) \\ internally chooses B = 4 %3 = [] ? bestapprPade(U, 5) \\ with B = 5, a solution exists %4 = (2*x^4 + x^3 - x - 1)/(-x^5 + x^3 + x^2 - 1)
* If x is a t_SER, we implicitly
convert the input to a t_POLMOD modulo N = tk where k is the
series absolute precision.
? T = 1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + O(t^7); \\ mod t^7 ? bestapprPade(T) %1 = 1/(-t + 1)
* If x is a t_SER and both B and Q are nonnegative,
returns a rational function approximation a/b
to x, with a of degree at most B and b of degree at most Q, with
x-a/b = O(tB+Q+1+v) if t is the variable, where v is the valuation
of x, the empty vector if not possible.
* If x is a t_RFRAC, we implicitly convert the input to a
t_POLMOD modulo N = tk where k = 2B + 1. If B was omitted,
we return x:
? T = (4*t^2 + 2*t + 3)/(t+1)^10; ? bestapprPade(T,1) %2 = [] \\ impossible ? bestapprPade(T,2) %3 = 27/(337*t^2 + 84*t + 9) ? bestapprPade(T,3) %4 = (4253*t - 3345)/(-39007*t^3 - 28519*t^2 - 8989*t - 1115)
The function applies recursively to components of complex objects (polynomials, vectors,...). If rational reconstruction fails for even a single entry, return [].
The library syntax is GEN bestapprPade0(GEN x, long B, long Q).
GEN bestapprPade(GEN x, long B) as bestapprPade0 when Q is ommited.
Deprecated alias for gcdext
The library syntax is GEN gcdext0(GEN x, GEN y).
Number of prime divisors of the integer |x| counted with multiplicity:
? factor(392) %1 = [2 3] [7 2] ? bigomega(392) %2 = 5; \\ = 3+2 ? omega(392) %3 = 2; \\ without multiplicity
The library syntax is long bigomega(GEN x).
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (dj) represents ∑j ≤ k ℤ/djℤ with dk
| ... | d1; any object which has a .cyc method is also
allowed, e.g. the output of znstar or bnrinit. A character
on this group is given by a row vector χ = [a1,...,an] such that
χ(∏ gjnj) = exp(2π i∑j aj nj / dj), where
gj denotes the generator (of order dj) of the j-th cyclic
component. This function returns the conjugate character.
? cyc = [15,5]; chi = [1,1]; ? charconj(cyc, chi) %2 = [14, 4] ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? charconj(bnf, [1]) %5 = [2]
For Dirichlet characters (when cyc is
znstar(q,1)), characters in Conrey representation are available,
see Section se:dirichletchar or ??character:
? G = znstar(16, 1); \\ (Z/16Z)* ? charconj(G, 3) \\ Conrey label %2 = [1, 1]~ ? znconreyexp(G, %) %3 = 11 \\ attached Conrey label; indeed 11 = 3^(-1) mod 16 ? chi = znconreylog(G, 3); ? charconj(G, chi) \\ Conrey logarithm %5 = [1, 1]~
The library syntax is GEN charconj0(GEN cyc, GEN chi).
Also available is
GEN charconj(GEN cyc, GEN chi), when cyc is known to
be a vector of elementary divisors and chi a compatible character
(no checks).
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (dj) represents ∑j ≤ k ℤ/djℤ with dk.
| ... | d1; any object which has a .cyc method is also
allowed, e.g. the output of znstar or bnrinit. A character
on this group is given by a row vector a = [a1,...,an] such that
χ(∏ gjnj) = exp(2π i∑j aj nj / dj),
where gj denotes the generator (of order dj) of the j-th cyclic
component.
Given two characters a and b, return the character a / b = a b.
? cyc = [15,5]; a = [1,1]; b = [2,4]; ? chardiv(cyc, a,b) %2 = [14, 2] ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? chardiv(bnf, [1], [2]) %5 = [2]
For Dirichlet characters on (ℤ/Nℤ)*, additional
representations are available (Conrey labels, Conrey logarithm),
see Section se:dirichletchar or ??character.
If the two characters are in the same format, the
result is given in the same format, otherwise a Conrey logarithm is used.
? G = znstar(100, 1); ? G.cyc %2 = [20, 2] ? a = [10, 1]; \\ usual representation for characters ? b = 7; \\ Conrey label; ? c = znconreylog(G, 11); \\ Conrey log ? chardiv(G, b,b) %6 = 1 \\ Conrey label ? chardiv(G, a,b) %7 = [0, 5]~ \\ Conrey log ? chardiv(G, a,c) %7 = [0, 14]~ \\ Conrey log
The library syntax is GEN chardiv0(GEN cyc, GEN a, GEN b).
Also available is
GEN chardiv(GEN cyc, GEN a, GEN b), when cyc is known to
be a vector of elementary divisors and a, b are compatible characters
(no checks).
Let G be an abelian group structure affording a discrete logarithm
method, e.g G = znstar(N, 1) for (ℤ/Nℤ)* or a bnr
structure, let x be an element of G and let chi be a character of
G (see the note below for details). This function returns the value of
chi at x.
Note on characters. Let K be some field. If G is an abelian group, let χ: G → K* be a character of finite order and let o be a multiple of the character order such that χ(n) = ζc(n) for some fixed ζ ∈ K* of multiplicative order o and a unique morphism c: G → (ℤ/oℤ,+). Our usual convention is to write G = (ℤ/o1ℤ) g1 ⨁ ...⨁ (ℤ/odℤ) gd for some generators (gi) of respective order di, where the group has exponent o := lcmi oi. Since ζo = 1, the vector (ci) in ∏i (ℤ/oiℤ) defines a character χ on G via χ(gi) = ζci (o/oi) for all i. Classical Dirichlet characters have values in K = ℂ and we can take ζ = exp(2iπ/o).
Note on Dirichlet characters.
In the special case where bid is attached to G = (ℤ/qℤ)*
(as per G = znstar(q,1)), the Dirichlet
character chi can be written in one of the usual 3 formats: a t_VEC
in terms of bid.gen as above, a t_COL in terms of the Conrey
generators, or a t_INT (Conrey label);
see Section se:dirichletchar or ??character.
The character value is encoded as follows, depending on the optional argument z:
* If z is omitted: return the rational number c(x)/o for x coprime to q, where we normalize 0 ≤ c(x) < o. If x can not be mapped to the group (e.g. x is not coprime to the conductor of a Dirichlet or Hecke character) we return the sentinel value -1.
* If z is an integer o, then we assume that o is a multiple of the character order and we return the integer c(x) when x belongs to the group, and the sentinel value -1 otherwise.
* z can be of the form [zeta, o], where zeta is an o-th root of 1 and o is a multiple of the character order. We return ζc(x) if x belongs to the group, and the sentinel value 0 otherwise. (Note that this coincides with the usual extension of Dirichlet characters to ℤ, or of Hecke characters to general ideals.)
* Finally, z can be of the form [vzeta, o], where vzeta is a vector of powers ζ0,..., ζo-1 of some o-th root of 1 and o is a multiple of the character order. As above, we return ζc(x) after a table lookup. Or the sentinel value 0.
The library syntax is GEN chareval(GEN G, GEN chi, GEN x, GEN z = NULL).
Let cyc represent a finite abelian group by its elementary divisors
(any object which has a .cyc method is also allowed, i.e. the output of
znstar or bnrinit). Return a list of representatives for the
Galois orbits of complex characters of G.
If ORD is present, select characters depending on their orders:
* if ORD is a t_INT, restrict to orders less than this
bound;
* if ORD is a t_VEC or t_VECSMALL, restrict to orders in
the list.
? G = znstar(96); ? #chargalois(G) \\ 16 orbits of characters mod 96 %2 = 16 ? #chargalois(G,4) \\ order less than 4 %3 = 12 ? chargalois(G,[1,4]) \\ order 1 or 4; 5 orbits %4 = [[0, 0, 0], [2, 0, 0], [2, 1, 0], [2, 0, 1], [2, 1, 1]]
Given a character χ, of order n (charorder(G,chi)), the
elements in its orbit are the φ(n) characters χi, (i,n) = 1.
The library syntax is GEN chargalois(GEN cyc, GEN ORD = NULL).
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (dj) represents ∑j ≤ k ℤ/djℤ with dk
| ... | d1; any object which has a .cyc method is also
allowed, e.g. the output of znstar or bnrinit. A character
on this group is given by a row vector χ = [a1,...,an] such that
χ(∏ gjnj) = exp(2π i∑j aj nj / dj),
where gj denotes
the generator (of order dj) of the j-th cyclic component.
This function returns the kernel of χ, as a matrix K in HNF which is a
left-divisor of matdiagonal(d). Its columns express in terms of
the gj the generators of the subgroup. The determinant of K is the
kernel index.
? cyc = [15,5]; chi = [1,1]; ? charker(cyc, chi) %2 = [15 12] [ 0 1] ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? charker(bnf, [1]) %5 = [3]
Note that for Dirichlet characters (when cyc is
znstar(q, 1)), characters in Conrey representation are available,
see Section se:dirichletchar or ??character.
? G = znstar(8, 1); \\ (Z/8Z)* ? charker(G, 1) \\ Conrey label for trivial character %2 = [1 0] [0 1]
The library syntax is GEN charker0(GEN cyc, GEN chi).
Also available is
GEN charker(GEN cyc, GEN chi), when cyc is known to
be a vector of elementary divisors and chi a compatible character
(no checks).
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (dj) represents ∑j ≤ k ℤ/djℤ with dk
| ... | d1; any object which has a .cyc method is also
allowed, e.g. the output of znstar or bnrinit. A character
on this group is given by a row vector χ = [a1,...,an] such that
χ(∏ gjnj) = exp(2π i∑j aj nj / dj),
where gj denotes
the generator (of order dj) of the j-th cyclic component.
Given two characters a and b, return the product character ab.
? cyc = [15,5]; a = [1,1]; b = [2,4]; ? charmul(cyc, a,b) %2 = [3, 0] ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? charmul(bnf, [1], [2]) %5 = [0]
For Dirichlet characters on (ℤ/Nℤ)*, additional
representations are available (Conrey labels, Conrey logarithm), see
Section se:dirichletchar or ??character. If the two characters are in
the same format, their
product is given in the same format, otherwise a Conrey logarithm is used.
? G = znstar(100, 1); ? G.cyc %2 = [20, 2] ? a = [10, 1]; \\ usual representation for characters ? b = 7; \\ Conrey label; ? c = znconreylog(G, 11); \\ Conrey log ? charmul(G, b,b) %6 = 49 \\ Conrey label ? charmul(G, a,b) %7 = [0, 15]~ \\ Conrey log ? charmul(G, a,c) %7 = [0, 6]~ \\ Conrey log
The library syntax is GEN charmul0(GEN cyc, GEN a, GEN b).
Also available is
GEN charmul(GEN cyc, GEN a, GEN b), when cyc is known to
be a vector of elementary divisors and a, b are compatible characters
(no checks).
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (dj) represents ∑j ≤ k ℤ/djℤ with dk
| ... | d1; any object which has a .cyc method is also
allowed, e.g. the output of znstar or bnrinit. A character
on this group is given by a row vector χ = [a1,...,an] such that
χ(∏ gjnj) = exp(2π i∑j aj nj / dj),
where gj denotes
the generator (of order dj) of the j-th cyclic component.
This function returns the order of the character chi.
? cyc = [15,5]; chi = [1,1]; ? charorder(cyc, chi) %2 = 15 ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? charorder(bnf, [1]) %5 = 3
For Dirichlet characters (when cyc is
znstar(q, 1)), characters in Conrey representation are available,
see Section se:dirichletchar or ??character:
? G = znstar(100, 1); \\ (Z/100Z)* ? charorder(G, 7) \\ Conrey label %2 = 4
The library syntax is GEN charorder0(GEN cyc, GEN chi).
Also available is
GEN charorder(GEN cyc, GEN chi), when cyc is known to
be a vector of elementary divisors and chi a compatible character
(no checks).
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (dj) represents ∑j ≤ k ℤ/djℤ with dk
| ... | d1; any object which has a .cyc method is also
allowed, e.g. the output of znstar or bnrinit. A character
on this group is given by a row vector χ = [a1,...,an] such that
χ(∏ gjnj) = exp(2π i∑j aj nj / dj),
where gj denotes
the generator (of order dj) of the j-th cyclic component.
Given n ∈ ℤ and a character a, return the character an.
? cyc = [15,5]; a = [1,1]; ? charpow(cyc, a, 3) %2 = [3, 3] ? charpow(cyc, a, 5) %2 = [5, 0] ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? charpow(bnf, [1], 3) %5 = [0]
For Dirichlet characters on (ℤ/Nℤ)*, additional
representations are available (Conrey labels, Conrey logarithm), see
Section se:dirichletchar or ??character and the output uses the
same format as the input.
? G = znstar(100, 1); ? G.cyc %2 = [20, 2] ? a = [10, 1]; \\ standard representation for characters ? b = 7; \\ Conrey label; ? c = znconreylog(G, 11); \\ Conrey log ? charpow(G, a,3) %6 = [10, 1] \\ standard representation ? charpow(G, b,3) %7 = 43 \\ Conrey label ? charpow(G, c,3) %8 = [1, 8]~ \\ Conrey log
The library syntax is GEN charpow0(GEN cyc, GEN a, GEN n).
Also available is
GEN charpow(GEN cyc, GEN a, GEN n), when cyc is known to
be a vector of elementary divisors (no check).
If x and y are both intmods or both polmods, creates (with the same type) a z in the same residue class as x and in the same residue class as y, if it is possible.
? chinese(Mod(1,2), Mod(2,3)) %1 = Mod(5, 6) ? chinese(Mod(x,x^2-1), Mod(x+1,x^2+1)) %2 = Mod(-1/2*x^2 + x + 1/2, x^4 - 1)
This function also allows vector and matrix arguments, in which case the operation is recursively applied to each component of the vector or matrix.
? chinese([Mod(1,2),Mod(1,3)], [Mod(1,5),Mod(2,7)]) %3 = [Mod(1, 10), Mod(16, 21)]
For polynomial arguments in the same variable, the function is applied to each
coefficient; if the polynomials have different degrees, the high degree terms
are copied verbatim in the result, as if the missing high degree terms in the
polynomial of lowest degree had been Mod(0,1). Since the latter
behavior is usually not the desired one, we propose to convert the
polynomials to vectors of the same length first:
? P = x+1; Q = x^2+2*x+1; ? chinese(P*Mod(1,2), Q*Mod(1,3)) %4 = Mod(1, 3)*x^2 + Mod(5, 6)*x + Mod(3, 6) ? chinese(Vec(P,3)*Mod(1,2), Vec(Q,3)*Mod(1,3)) %5 = [Mod(1, 6), Mod(5, 6), Mod(4, 6)] ? Pol(%) %6 = Mod(1, 6)*x^2 + Mod(5, 6)*x + Mod(4, 6)
If y is omitted, and x is a vector, chinese is applied recursively
to the components of x, yielding a residue belonging to the same class as all
components of x.
Finally chinese(x,x) = x regardless of the type of x; this allows
vector arguments to contain other data, so long as they are identical in both
vectors.
The library syntax is GEN chinese(GEN x, GEN y = NULL).
GEN chinese1(GEN x) is also available.
Computes the gcd of all the coefficients of x, when this gcd makes sense. This is the natural definition if x is a polynomial (and by extension a power series) or a vector/matrix. This is in general a weaker notion than the ideal generated by the coefficients:
? content(2*x+y) %1 = 1 \\ = gcd(2,y) over Q[y]
If x is a scalar, this simply returns the absolute value of x if x is
rational (t_INT or t_FRAC), and either 1 (inexact input) or x
(exact input) otherwise; the result should be identical to gcd(x, 0).
The content of a rational function is the ratio of the contents of the numerator and the denominator. In recursive structures, if a matrix or vector coefficient x appears, the gcd is taken not with x, but with its content:
? content([ [2], 4*matid(3) ]) %1 = 2
The content of a t_VECSMALL is computed assuming the
entries are signed integers.
The optional argument D allows to control over which ring we compute and get a more predictable behaviour:
* 1: we only consider the underlying ℚ-structure and the denominator is a (positive) rational number
* a simple variable, say 'x: all entries are considered as
rational functions in K(x) for some field K and the content is an
element of K.
? f = x + 1/y + 1/2; ? content(f) \\ as a t_POL in x %2 = 1/(2*y) ? content(f, 1) \\ Q-content %3 = 1/2 ? content(f, y) \\ as a rational function in y %4 = 1/2 ? g = x^2*y + y^2*x; ? content(g, x) %6 = y ? content(g, y) %7 = x
The library syntax is GEN content0(GEN x, GEN D = NULL).
Returns the row vector whose components are the partial quotients of the continued fraction expansion of x. In other words, a result [a0,...,an] means that x ~ a0+1/(a1+...+1/an). The output is normalized so that an ! = 1 (unless we also have n = 0).
The number of partial quotients n+1 is limited by nmax. If
nmax is omitted, the expansion stops at the last significant partial
quotient.
? \p19
realprecision = 19 significant digits
? contfrac(Pi)
%1 = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2]
? contfrac(Pi,, 3) \\ n = 2
%2 = [3, 7, 15]
x can also be a rational function or a power series.
If a vector b is supplied, the numerators are equal to the coefficients
of b, instead of all equal to 1 as above; more precisely, x ~
(1/b0)(a0+b1/(a1+...+bn/an)); for a numerical continued
fraction (x real), the ai are integers, as large as possible;
if x is a
rational function, they are polynomials with deg ai = deg bi + 1.
The length of the result is then equal to the length of b, unless the next
partial quotient cannot be reliably computed, in which case the expansion
stops. This happens when a partial remainder is equal to zero (or too small
compared to the available significant digits for x a t_REAL).
A direct implementation of the numerical continued fraction
contfrac(x,b) described above would be
\\ "greedy" generalized continued fraction
cf(x, b) =
{ my( a= vector(#b), t );
x *= b[1];
for (i = 1, #b,
a[i] = floor(x);
t = x - a[i]; if (!t || i == #b, break);
x = b[i+1] / t;
); a;
}
There is some degree of freedom when choosing the ai; the program above can easily be modified to derive variants of the standard algorithm. In the same vein, although no builtin function implements the related Engel expansion (a special kind of Egyptian fraction decomposition: x = 1/a1 + 1/(a1a2) +...), it can be obtained as follows:
\\ n terms of the Engel expansion of x
engel(x, n = 10) =
{ my( u = x, a = vector(n) );
for (k = 1, n,
a[k] = ceil(1/u);
u = u*a[k] - 1;
if (!u, break);
); a
}
Obsolete hack. (don't use this): if b is an integer, nmax
is ignored and the command is understood as contfrac(x,, b).
The library syntax is GEN contfrac0(GEN x, GEN b = NULL, long nmax).
Also available are GEN gboundcf(GEN x, long nmax),
GEN gcf(GEN x) and GEN gcf2(GEN b, GEN x).
When x is a vector or a one-row matrix, x is considered as the list of partial quotients [a0,a1,...,an] of a rational number, and the result is the 2 by 2 matrix [pn,pn-1;qn,qn-1] in the standard notation of continued fractions, so pn/qn = a0+1/(a1+...+1/an). If x is a matrix with two rows [b0,b1,...,bn] and [a0,a1,...,an], this is then considered as a generalized continued fraction and we have similarly pn/qn = (1/b0)(a0+b1/(a1+...+bn/an)). Note that in this case one usually has b0 = 1.
If n ≥ 0 is present, returns all convergents from p0/q0 up to pn/qn. (All convergents if x is too small to compute the n+1 requested convergents.)
? a = contfrac(Pi,10) %1 = [3, 7, 15, 1, 292, 1, 1, 1, 3] ? allpnqn(x) = contfracpnqn(x,#x) \\ all convergents ? allpnqn(a) %3 = [3 22 333 355 103993 104348 208341 312689 1146408] [1 7 106 113 33102 33215 66317 99532 364913] ? contfracpnqn(a) \\ last two convergents %4 = [1146408 312689] [ 364913 99532] ? contfracpnqn(a,3) \\ first three convergents %5 = [3 22 333 355] [1 7 106 113]
The library syntax is GEN contfracpnqn(GEN x, long n).
also available is GEN pnqn(GEN x) for n = -1.
If n is an integer written as
n = df2 with d squarefree, returns d. If flag is nonzero,
returns the two-element row vector [d,f]. By convention, we write 0 = 0
x 12, so core(0, 1) returns [0,1].
The library syntax is GEN core0(GEN n, long flag).
Also available are GEN core(GEN n) (flag = 0) and
GEN core2(GEN n) (flag = 1)
A fundamental discriminant is an integer of the form t = 1 mod 4 or 4t = 8,12 mod 16, with t squarefree (i.e. 1 or the discriminant of a quadratic number field). Given a nonzero integer n, this routine returns the (unique) fundamental discriminant d such that n = df2, f a positive rational number. If flag is nonzero, returns the two-element row vector [d,f]. If n is congruent to 0 or 1 modulo 4, f is an integer, and a half-integer otherwise.
By convention, coredisc(0, 1)) returns [0,1].
Note that quaddisc(n) returns the same value as coredisc(n),
and also works with rational inputs n ∈ ℚ*.
The library syntax is GEN coredisc0(GEN n, long flag).
Also available are GEN coredisc(GEN n) (flag = 0) and
GEN coredisc2(GEN n) (flag = 1)
x and y being vectors of perhaps different lengths but with y[1] ! = 0 considered as Dirichlet series, computes the quotient of x by y, again as a vector.
The library syntax is GEN dirdiv(GEN x, GEN y).
Computes the Dirichlet series attached to the Euler product of expression expr as p ranges through the primes from a to b. expr must be a polynomial or rational function in another variable than p (say X) and expr(X) is understood as the local factor expr(p-s).
The series is output as a vector of coefficients. If c is omitted, output the first b coefficients of the series; otherwise, output the first c coefficients. The following command computes the sigma function, attached to ζ(s)ζ(s-1):
? direuler(p=2, 10, 1/((1-X)*(1-p*X))) %1 = [1, 3, 4, 7, 6, 12, 8, 15, 13, 18] ? direuler(p=2, 10, 1/((1-X)*(1-p*X)), 5) \\ fewer terms %2 = [1, 3, 4, 7, 6]
Setting c < b is useless (the same effect would be achieved by setting b = c). If c > b, the computed coefficients are "missing" Euler factors:
? direuler(p=2, 10, 1/((1-X)*(1-p*X)), 15) \\ more terms, no longer = sigma ! %3 = [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 0, 28, 0, 24, 24]
The library syntax is direuler(void *E, GEN (*eval)(void*,GEN), GEN a, GEN b)
x and y being vectors of perhaps different lengths representing the Dirichlet series ∑n xn n-s and ∑n yn n-s, computes the product of x by y, again as a vector.
? dirmul(vector(10,n,1), vector(10,n,moebius(n))) %1 = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
The product
length is the minimum of #x*v(y) and #y*v(x),
where v(x) is the index of the first nonzero coefficient.
? dirmul([0,1], [0,1]); %2 = [0, 0, 0, 1]
The library syntax is GEN dirmul(GEN x, GEN y).
For positive integer N and complex number x, return the sum
f(1)1× + f(2)2× +...+ f(N)N×, where f is a completely
multiplicative function. If f is omitted, return
1× +...+ N×. When N ≤ 0, the function returns 0.
If both is set, return the pair for arguments (x,f) and
(-1-x,f). If both = 2, assume in addition that f is
real-valued (which is true when f is omitted, i.e. represents the constant
function f(n) = 1).
Caveat. when both is set, the present implementation assumes that |f(n)| is either 0 or 1, which is the case for Dirichlet characters.
A vector-valued multiplicative function f is allowed, in which case the above conditions must be met componentwise and the vector length must be constant.
Unlike variants using dirpowers(N,x), this function uses O(sqrt{N})
memory instead of O(N). And it is faster for large N. The return value
is usually a floating point number, but it will be exact if the result
is an integer. On the other hand, rational numbers are converted to
floating point approximations, since they are likely to blow up for large N.
? dirpowers(5, 2) %1 = [1, 4, 9, 16, 25] ? vecsum(%) %2 = 55 ? dirpowerssum(5, 2) %3 = 55 ? dirpowerssum(5, -2) %4 = 1.4636111111111111111111111111111111111 ? \p200 ? s = 1/2 + I * sqrt(3); N = 10^7; ? dirpowerssum(N, s); time = 11,425 ms. ? vecsum(dirpowers(N, s)) time = 19,365 ms. ? dirpowerssum(N, s, n->kronecker(-23,n)) time = 10,981 ms.
The dirpowerssum commands work with default stack size,
the dirpowers one requires a stacksize of at least 5GB.
The library syntax is dirpowerssumfun(ulong N, GEN x, void *E, GEN (*f)(void*, ulong, long), long prec). When f = NULL, one may use
GEN dirpowerssum(ulong N, GEN x, long prec).
Creates a row vector whose components are the
divisors of x. The factorization of x (as output by factor) can
be used instead. If flag = 1, return pairs [d, factor(d)].
By definition, these divisors are the products of the irreducible
factors of n, as produced by factor(n), raised to appropriate
powers (no negative exponent may occur in the factorization). If n is
an integer, they are the positive divisors, in increasing order.
? divisors(12)
%1 = [1, 2, 3, 4, 6, 12]
? divisors(12, 1) \\ include their factorization
%2 = [[1, matrix(0,2)], [2, Mat([2, 1])], [3, Mat([3, 1])],
[4, Mat([2, 2])], [6, [2, 1; 3, 1]], [12, [2, 2; 3, 1]]]
? divisors(x^4 + 2*x^3 + x^2) \\ also works for polynomials
%3 = [1, x, x^2, x + 1, x^2 + x, x^3 + x^2, x^2 + 2*x + 1,
x^3 + 2*x^2 + x, x^4 + 2*x^3 + x^2]
This function requires a lot of memory if x has many divisors. The following idiom runs through all divisors using very little memory, in no particular order this time:
F = factor(x); P = F[,1]; E = F[,2]; forvec(e = vectorv(#E,i,[0,E[i]]), d = factorback(P,e); ...)
If the factorization of d is also desired, then [P,e] almost provides it but not quite: e may contain 0 exponents, which are not allowed in factorizations. These must be sieved out as in:
? tofact(P,E) = matreduce(Mat([P,E])); ? tofact([2,3,5,7]~, [4,0,2,0]~) %4 = [2 4] [5 2]
We can then run the above loop with tofact(P,e) instead of,
or together with, factorback.
The library syntax is GEN divisors0(GEN x, long flag).
The functions GEN divisors(GEN N) (flag = 0) and
GEN divisors_factored(GEN N) (flag = 1) are also available.
Given three integers N > s > r ≥ 0 such that (r,s) = 1 and s3 > N, find all divisors d of N such that d = r (mod s). There are at most 11 such divisors (Lenstra).
? N = 245784; r = 19; s = 65 ; ? divisorslenstra(N, r, s) %2 = [19, 84, 539, 1254, 3724, 245784] ? [ d | d <- divisors(N), d % s == r] %3 = [19, 84, 539, 1254, 3724, 245784]
When the preconditions are not met, the result is undefined:
? N = 4484075232; r = 7; s = 1303; s^3 > N %4 = 0 ? divisorslenstra(N, r, s) ? [ d | d <- divisors(N), d % s == r ] %6 = [7, 2613, 9128, 19552, 264516, 3407352, 344928864]
(Divisors were missing but s3 < N.)
The library syntax is GEN divisorslenstra(GEN N, GEN r, GEN s).
Euler's φ (totient) function of the integer |x|, in other words |(ℤ/xℤ)*|.
? eulerphi(40) %1 = 16
According to this definition we let φ(0) := 2, since ℤ* = {-1,1};
this is consistent with znstar(0): we have
znstar(n).no = eulerphi(n) for all n ∈ ℤ.
The library syntax is GEN eulerphi(GEN x).
Factor x over domain D; if D is omitted, it is determined from x. For instance, if x is an integer, it is factored in ℤ, if it is a polynomial with rational coefficients, it is factored in ℚ[x], etc., see below for details. The result is a two-column matrix: the first contains the irreducibles dividing x (rational or Gaussian primes, irreducible polynomials), and the second the exponents. By convention, 0 is factored as 01.
x ∈ ℚ.
See factorint for the algorithms used. The factorization includes the
unit -1 when x < 0 and all other factors are positive; a denominator is
factored with negative exponents. The factors are sorted in increasing order.
? factor(-7/106) %1 = [-1 1] [ 2 -1] [ 7 1] [53 -1]
By convention, 1 is factored as matrix(0,2)
(the empty factorization, printed as [;]).
Large rational "primes" > 264 in the factorization are in fact
pseudoprimes (see ispseudoprime), a priori not rigorously proven
primes. Use isprime to prove primality of these factors, as in
? fa = factor(2^2^7 + 1) %2 = [59649589127497217 1] [5704689200685129054721 1] ? isprime( fa[,1] ) %3 = [1, 1]~ \\ both entries are proven primes
Another possibility is to globally set the default factor_proven, which
will perform a rigorous primality proof for each pseudoprime factor but will
slow down PARI.
A t_INT argument D can be added, meaning that we only trial divide
by all primes p < D and the addprimes entries, then skip all
expensive factorization methods. The limit D must be nonnegative.
In this case, one entry in the factorization may be a composite number: all
factors less than D2 and primes from the addprimes table
are actual primes. But (at most) one entry may not verify this criterion,
and it may be prime or composite: it is only known to be coprime to all
other entries and not a pure power.
? factor(2^2^7 +1, 10^5) %4 = [340282366920938463463374607431768211457 1]
Deprecated feature. Setting D = 0 is the same
as setting it to factorlimit + 1.
This routine uses trial division and perfect power tests, and should not be
used for huge values of D (at most 109, say):
factorint(, 1 + 8) will in general be faster. The latter does not
guarantee that all small prime factors are found, but it also finds larger
factors and in a more efficient way.
? F = (2^2^7 + 1) * 1009 * (10^5+3); factor(F, 10^5) \\ fast, incomplete time = 0 ms. %5 = [1009 1] [34029257539194609161727850866999116450334371 1] ? factor(F, 10^9) \\ slow time = 3,260 ms. %6 = [1009 1] [100003 1] [340282366920938463463374607431768211457 1] ? factorint(F, 1+8) \\ much faster and all small primes were found time = 8 ms. %7 = [1009 1] [100003 1] [340282366920938463463374607431768211457 1] ? factor(F) \\ complete factorization time = 60 ms. %8 = [1009 1] [100003 1] [59649589127497217 1] [5704689200685129054721 1]
x ∈ ℚ(i). The factorization is performed with Gaussian primes in ℤ[i] and includes Gaussian units in {±1, ± i}; factors are sorted by increasing norm. Except for a possible leading unit, the Gaussian factors are normalized: rational factors are positive and irrational factors have positive imaginary part.
Unless factor_proven is set, large factors are actually pseudoprimes,
not proven primes; a rational factor is prime if less than 264 and an
irrational one if its norm is less than 264.
? factor(5*I) %9 = [ 2 + I 1] [1 + 2*I 1]
One can force the factorization of a rational number by setting the domain D = I:
? factor(-5, I) %10 = [ I 1] [ 2 + I 1] [1 + 2*I 1] ? factorback(%) %11 = -5
Univariate polynomials and rational functions.
PARI can factor univariate polynomials in K[t]. The following base fields
K are currently supported: ℚ, ℝ, ℂ, ℚp, finite fields and
number fields. See factormod and factorff for the algorithms used
over finite fields and nffactor for the algorithms over number fields.
The irreducible factors are sorted by increasing degree and normalized: they
are monic except when K = ℚ where they are primitive in ℤ[t].
The content is not included in the factorization, in particular
factorback will in general recover the original x only up to
multiplication by an element of K*: when K ! = ℚ, this scalar is
pollead(x) (since irreducible factors are monic); and when K = ℚ
you can either ask for the ℚ-content explicitly of use factorback:
? P = t^2 + 5*t/2 + 1; F = factor(P) %12 = [t + 2 1] [2*t + 1 1] ? content(P, 1) \\ Q-content %13 = 1/2 ? pollead(factorback(F)) / pollead(P) %14 = 2
You can specify K using the optional "domain" argument D as follows
* K = ℚ : D a rational number (t_INT or t_FRAC),
* K = ℤ/pℤ with p prime : D a t_INTMOD modulo p;
factoring modulo a composite number is not supported.
* K = 𝔽q : D a t_FFELT encoding the finite field; you can also
use a t_POLMOD of t_INTMOD modulo a prime p but this is usualy
less convenient;
* K = ℚ[X]/(T) a number field : D a t_POLMOD modulo T,
* K = ℚ(i) (alternate syntax for special case): D = I,
* K = ℚ(w) a quadratic number field (alternate syntax for special
case): D a t_QUAD,
* K = ℝ : D a real number (t_REAL); truncate the factorization
at accuracy precision(D). If x is inexact and precision(x)
is less than precision(D), then the precision of x is used instead.
* K = ℂ : D a complex number with a t_REAL component, e.g.
I * 1.; truncate the factorization as for K = ℝ,
* K = ℚp : D a t_PADIC; truncate the factorization at
p-adic accuracy padicprec(D), possibly less if x is inexact
with insufficient p-adic accuracy;
? T = x^2+1; ? factor(T, 1); \\ over Q ? factor(T, Mod(1,3)) \\ over F3 ? factor(T, ffgen(ffinit(3,2,'t))^0) \\ over F3^2 ? factor(T, Mod(Mod(1,3), t^2+t+2)) \\ over F3^2, again ? factor(T, O(3^6)) \\ over Q3, precision 6 ? factor(T, 1.) \\ over R, current precision ? factor(T, I*1.) \\ over C ? factor(T, Mod(1, y^3-2)) \\ over Q(21/3)
In most cases, it is possible and simpler to call a specialized variant rather than use the above scheme:
? factormod(T, 3) \\ over F3 ? factormod(T, [t^2+t+2, 3]) \\ over F3^2 ? factormod(T, ffgen(3^2, 't)) \\ over F3^2 ? factorpadic(T, 3,6) \\ over Q3, precision 6 ? nffactor(y^3-2, T) \\ over Q(21/3) ? polroots(T) \\ over C ? polrootsreal(T) \\ over R (real polynomial)
It is also possible to let the routine use the smallest field containing all
coefficients, taking into account quotient structures induced by
t_INTMODs and t_POLMODs (e.g. if a coefficient in ℤ/nℤ is known,
all rational numbers encountered are first mapped to ℤ/nℤ; different
moduli will produce an error):
? T = x^2+1; ? factor(T); \\ over Q ? factor(T*Mod(1,3)) \\ over F3 ? factor(T*ffgen(ffinit(3,2,'t))^0) \\ over F3^2 ? factor(T*Mod(Mod(1,3), t^2+t+2)) \\ over F3^2, again ? factor(T*(1 + O(3^6)) \\ over Q3, precision 6 ? factor(T*1.) \\ over R, current precision ? factor(T*(1.+0.*I)) \\ over C ? factor(T*Mod(1, y^3-2)) \\ over Q(21/3)
Multiplying by a suitable field element equal to 1 ∈ K in this way is error-prone and is not recommanded. Factoring existing polynomials with obvious fields of coefficients is fine, the domain argument D should be used instead ad hoc conversions.
Note on inexact polynomials. Polynomials with inexact coefficients (e.g. floating point or p-adic numbers) are first rounded to an exact representation, then factored to (potentially) infinite accuracy and we return a truncated approximation of that virtual factorization. To avoid pitfalls, we advise to only factor exact polynomials:
? factor(x^2-1+O(2^2)) \\ rounded to x^2 + 3, irreducible in Q2 %1 = [(1 + O(2^2))*x^2 + O(2^2)*x + (1 + 2 + O(2^2)) 1] ? factor(x^2-1+O(2^3)) \\ rounded to x^2 + 7, reducible ! %2 = [ (1 + O(2^3))*x + (1 + 2 + O(2^3)) 1] [(1 + O(2^3))*x + (1 + 2^2 + O(2^3)) 1] ? factor(x^2-1, O(2^2)) \\ no ambiguity now %3 = [ (1 + O(2^2))*x + (1 + O(2^2)) 1] [(1 + O(2^2))*x + (1 + 2 + O(2^2)) 1]
Note about inseparable polynomials. Polynomials with inexact
coefficients are considered to be squarefree: indeed, there exist a
squarefree polynomial arbitrarily close to the input, and they cannot be
distinguished at the input accuracy. This means that irreducible factors are
repeated according to their apparent multiplicity. On the contrary, using a
specialized function such as factorpadic with an exact rational
input yields the correct multiplicity when the (now exact) input is not
separable. Compare:
? factor(z^2 + O(5^2))) %1 = [(1 + O(5^2))*z + O(5^2) 1] [(1 + O(5^2))*z + O(5^2) 1] ? factor(z^2, O(5^2)) %2 = [1 + O(5^2))*z + O(5^2) 2]
Multivariate polynomials and rational functions. PARI recursively factors multivariate polynomials in K[t1,..., td] for the same fields K as above and the argument D is used in the same way to specify K. The irreducible factors are sorted by their main variable (least priority first) then by increasing degree.
? factor(x^2 + y^2, Mod(1,5)) %1 = [ x + Mod(2, 5)*y 1] [Mod(1, 5)*x + Mod(3, 5)*y 1] ? factor(x^2 + y^2, O(5^2)) %2 = [ (1 + O(5^2))*x + (O(5^2)*y^2 + (2 + 5 + O(5^2))*y + O(5^2)) 1] [(1 + O(5^2))*x + (O(5^2)*y^2 + (3 + 3*5 + O(5^2))*y + O(5^2)) 1] ? lift(%) %3 = [ x + 7*y 1] [x + 18*y 1]
Note that the implementation does not really support inexact real fields (ℝ or ℂ) and usually misses factors even if the input is exact:
? factor(x^2 + y^2, I) \\ over Q(i) %4 = [x - I*y 1] [x + I*y 1] ? factor(x^2 + y^2, I*1.) \\ over C %5 = [x^2 + y^2 1]
The library syntax is GEN factor0(GEN x, GEN D = NULL).
GEN factor(GEN x)
GEN boundfact(GEN x, ulong lim).
Gives back the factored object corresponding to a factorization. The integer 1 corresponds to the empty factorization.
If e is present, e and f must be vectors of the same length (e being integral), and the corresponding factorization is the product of the f[i]e[i].
If not, and f is vector, it is understood as in the preceding case with e
a vector of 1s: we return the product of the f[i]. Finally, f can be a
regular factorization, as produced with any factor command. A few
examples:
? factor(12) %1 = [2 2] [3 1] ? factorback(%) %2 = 12 ? factorback([2,3], [2,1]) \\ 2^2 * 3^1 %3 = 12 ? factorback([5,2,3]) %4 = 30
The library syntax is GEN factorback2(GEN f, GEN e = NULL).
Also available is GEN factorback(GEN f) (case e = NULL).
This function is obsolete, use factormod.
The library syntax is GEN factmod(GEN x, GEN p).
Obsolete, kept for backward compatibility: use factormod.
The library syntax is GEN factorff(GEN x, GEN p = NULL, GEN a = NULL).
Factorial of x. The expression x! gives a result which is an integer,
while factorial(x) gives a real number.
The library syntax is GEN mpfactr(long x, long prec).
GEN mpfact(long x) returns x! as a t_INT.
Factors the integer n into a product of
pseudoprimes (see ispseudoprime), using a combination of the
Shanks SQUFOF and Pollard Rho method (with modifications due to
Brent), Lenstra's ECM (with modifications by Montgomery), and
MPQS (the latter adapted from the LiDIA code with the kind
permission of the LiDIA maintainers), as well as a search for pure powers.
The output is a two-column matrix as for factor: the first column
contains the "prime" divisors of n, the second one contains the
(positive) exponents.
By convention 0 is factored as 01, and 1 as the empty factorization;
also the divisors are by default not proven primes if they are larger than
264, they only failed the BPSW compositeness test (see
ispseudoprime). Use isprime on the result if you want to
guarantee primality or set the factor_proven default to 1.
Entries of the private prime tables (see addprimes) are also included
as is.
This gives direct access to the integer factoring engine called by most arithmetical functions. flag is optional; its binary digits mean 1: avoid MPQS, 2: skip first stage ECM (we may still fall back to it later), 4: avoid Rho and SQUFOF, 8: don't run final ECM (as a result, a huge composite may be declared to be prime). Note that a (strong) probabilistic primality test is used; thus composites might not be detected, although no example is known.
You are invited to play with the flag settings and watch the internals at
work by using gp's debug default parameter (level 3 shows
just the outline, 4 turns on time keeping, 5 and above show an increasing
amount of internal details).
The library syntax is GEN factorint(GEN x, long flag).
Factors the polynomial f over the finite field defined by the domain D as follows:
* D = p a prime: factor over 𝔽p;
* D = [T,p] for a prime p and T(y) an irreducible polynomial over 𝔽p: factor over 𝔽p[y]/(T) (as usual the main variable of T must have lower priority than the main variable of f);
* D a t_FFELT: factor over the attached field;
* D omitted: factor over the field of definition of f, which must be a finite field.
The coefficients of f must be operation-compatible with the corresponding finite field. The result is a two-column matrix, the first column being the irreducible polynomials dividing f, and the second the exponents. By convention, the 0 polynomial factors as 01; a nonzero constant polynomial has empty factorization, a 0 x 2 matrix. The irreducible factors are ordered by increasing degree and the result is canonical: it will not change across multiple calls or sessions.
? factormod(x^2 + 1, 3) \\ over F3 %1 = [Mod(1, 3)*x^2 + Mod(1, 3) 1] ? liftall( factormod(x^2 + 1, [t^2+1, 3]) ) \\ over F9 %2 = [ x + t 1] [x + 2*t 1] \\ same, now letting GP choose a model ? T = ffinit(3,2,'t) %3 = Mod(1, 3)*t^2 + Mod(1, 3)*t + Mod(2, 3) ? liftall( factormod(x^2 + 1, [T, 3]) ) %4 = \\ t is a root of T ! [ x + (t + 2) 1] [x + (2*t + 1) 1] ? t = ffgen(t^2+Mod(1,3)); factormod(x^2 + t^0) \\ same using t_FFELT %5 = [ x + t 1] [x + 2*t 1] ? factormod(x^2+Mod(1,3)) %6 = [Mod(1, 3)*x^2 + Mod(1, 3) 1] ? liftall( factormod(x^2 + Mod(Mod(1,3), y^2+1)) ) %7 = [ x + y 1] [x + 2*y 1]
If flag is nonzero, outputs only the degrees of the irreducible polynomials (for example to compute an L-function). By convention, a constant polynomial (including the 0 polynomial) has empty factorization. The degrees appear in increasing order but need not correspond to the ordering with flag = 0 when multiplicities are present.
? f = x^3 + 2*x^2 + x + 2; ? factormod(f, 5) \\ (x+2)^2 * (x+3) %1 = [Mod(1, 5)*x + Mod(2, 5) 2] [Mod(1, 5)*x + Mod(3, 5) 1] ? factormod(f, 5, 1) \\ (deg 1) * (deg 1)^2 %2 = [1 1] [1 2]
The library syntax is GEN factormod0(GEN f, GEN D = NULL, long flag).
Distinct-degree factorization of the squarefree polynomial f over the finite field defined by the domain D as follows:
* D = p a prime: factor over 𝔽p;
* D = [T,p] for a prime p and T an irreducible polynomial over 𝔽p: factor over 𝔽p[x]/(T);
* D a t_FFELT: factor over the attached field;
* D omitted: factor over the field of definition of f, which must be a finite field.
If f is not squarefree, the result is undefined. The coefficients of f must be operation-compatible with the corresponding finite field. The result is a two-column matrix:
* the first column contains monic (squarefree, pairwise coprime) polynomials dividing f, all of whose irreducible factors have the same degree d;
* the second column contains the degrees of the irreducible factors.
The factorization is ordered by increasing degree d of irreducible factors, and the result is obviously canonical. This function is somewhat faster than full factorization.
? f = (x^2 + 1) * (x^2-1); ? factormodSQF(f,3) \\ squarefree over F3 %2 = [Mod(1, 3)*x^4 + Mod(2, 3) 1] ? factormodDDF(f, 3) %3 = [Mod(1, 3)*x^2 + Mod(2, 3) 1] \\ two degree 1 factors [Mod(1, 3)*x^2 + Mod(1, 3) 2] \\ irred of degree 2 ? for(i=1,10^5,factormodDDF(f,3)) time = 424 ms. ? for(i=1,10^5,factormod(f,3)) \\ full factorization is a little slower time = 464 ms. ? liftall( factormodDDF(x^2 + 1, [3, t^2+1]) ) \\ over F9 %6 = [x^2 + 1 1] \\ product of two degree 1 factors ? t = ffgen(t^2+Mod(1,3)); factormodDDF(x^2 + t^0) \\ same using t_FFELT %7 = [x^2 + 1 1] ? factormodDDF(x^2-Mod(1,3)) %8 = [Mod(1, 3)*x^2 + Mod(2, 3) 1]
The library syntax is GEN factormodDDF(GEN f, GEN D = NULL).
Squarefree factorization of the polynomial f over the finite field defined by the domain D as follows:
* D = p a prime: factor over 𝔽p;
* D = [T,p] for a prime p and T an irreducible polynomial over 𝔽p: factor over 𝔽p[x]/(T);
* D a t_FFELT: factor over the attached field;
* D omitted: factor over the field of definition of f, which must be a finite field.
The coefficients of f must be operation-compatible with the corresponding finite field. The result is a two-column matrix:
* the first column contains monic squarefree pairwise coprime polynomials dividing f;
* the second column contains the power to which the polynomial in column 1 divides f;
This is somewhat faster than full factorization. The factors are ordered by increasing exponent and the result is obviously canonical.
? f = (x^2 + 1)^3 * (x^2-1)^2; ? factormodSQF(f, 3) \\ over F3 %1 = [Mod(1, 3)*x^2 + Mod(2, 3) 2] [Mod(1, 3)*x^2 + Mod(1, 3) 3] ? for(i=1,10^5,factormodSQF(f,3)) time = 192 ms. ? for(i=1,10^5,factormod(f,3)) \\ full factorization is slower time = 409 ms. ? liftall( factormodSQF((x^2 + 1)^3, [3, t^2+1]) ) \\ over F9 %4 = [x^2 + 1 3] ? t = ffgen(t^2+Mod(1,3)); factormodSQF((x^2 + t^0)^3) \\ same using t_FFELT %5 = [x^2 + 1 3] ? factormodSQF(x^8 + x^7 + x^6 + x^2 + x + Mod(1,2)) %6 = [ Mod(1, 2)*x + Mod(1, 2) 2] [Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2) 3]
The library syntax is GEN factormodSQF(GEN f, GEN D = NULL).
Factors n-th cyclotomic polynomial Φn(x) mod p,
where p is a prime number not dividing n.
Much faster than factormod(polcyclo(n), p); the irreducible
factors should be identical and given in the same order.
If single is set, return a single irreducible factor; else (default)
return all the irreducible factors. Note that repeated calls of this
function with the single flag set may return different results because
the algorithm is probabilistic. Algorithms used are as follows.
Let F = ℚ(ζn). Let K be the splitting field of p in F and e the conductor of K. Then Φn(x) and Φe(x) have the same number of irreducible factors mod p and there is a simple algorithm constructing irreducible factors of Φn(x) from irreducible factors of Φe(x). So we may assume n is equal to the conductor of K. Let d be the order of p in (ℤ/nℤ)× and ϕ(n) = df. Then Φn(x) has f irreducible factors gi(x) (1 ≤ i ≤ f) of degree d over 𝔽p or ℤp.
* If d is small, then we factor gi(x) into d linear factors gij(x), 1 ≤ j ≤ d in 𝔽q[x] (q = pd) and construct Gi(x) = ∏j = 1d gij(x) ∈ 𝔽q[x]. Then Gi(x) ∈ 𝔽p[x] and gi(x) = Gi(x).
* If f is small, then we work in K, which is a Galois extension of degree f over ℚ. The Gaussian period θk = TrF/K(ζnk) is a sum of k-th power of roots of gi(x) and K = ℚ(θ1).
Now, for each k, there is a polynomial Tk(x) ∈ ℚ[x] satisfying θk = Tk(θ1) because all θk are in K. Let T(x) ∈ ℤ[x] be the minimal polynomial of θ1 over ℚ. We get θ1 mod p from T(x) and construct θ1,...,θd mod p using Tk(x). Finally we recover gi(x) from θ1,...,θd by Newton's formula.
? lift(factormodcyclo(15, 11)) %1 = [x^2 + 9*x + 4, x^2 + 4*x + 5, x^2 + 3*x + 9, x^2 + 5*x + 3] ? factormodcyclo(15, 11, 1) \\ single %2 = Mod(1, 11)*x^2 + Mod(5, 11)*x + Mod(3, 11) ? z1 = lift(factormod(polcyclo(12345),11311)[,1]); time = 32,498 ms. ? z2 = factormodcyclo(12345,11311); time = 47 ms. ? z1 == z2 %4 = 1
The library syntax is GEN factormodcyclo(long n, GEN p, long single, long v = -1) where v is a variable number.
Let k, l, m be three finite fields and f a (partial) map from l to m and g a (partial) map from k to l, return the (partial) map f o g from k to m.
a = ffgen([3,5],'a); b = ffgen([3,10],'b); c = ffgen([3,20],'c); m = ffembed(a, b); n = ffembed(b, c); rm = ffinvmap(m); rn = ffinvmap(n); nm = ffcompomap(n,m); ffmap(n,ffmap(m,a)) == ffmap(nm, a) %5 = 1 ffcompomap(rm, rn) == ffinvmap(nm) %6 = 1
The library syntax is GEN ffcompomap(GEN f, GEN g).
Given two finite fields elements a and b, return a map embedding the definition field of a to the definition field of b. Assume that the latter contains the former.
? a = ffgen([3,5],'a); ? b = ffgen([3,10],'b); ? m = ffembed(a, b); ? A = ffmap(m, a); ? minpoly(A) == minpoly(a) %5 = 1
The library syntax is GEN ffembed(GEN a, GEN b).
Extend the field K of definition of a by a root of the polynomial
P ∈ K[X] assumed to be irreducible over K. Return [r, m] where r
is a root of P in the extension field L and m is a map from K to L,
see ffmap.
If v is given, the variable name is used to display the generator of L,
else the name of the variable of P is used.
A generator of L can be recovered using b = ffgen(r).
The image of P in L[X] can be recovered using PL = ffmap(m,P).
? a = ffgen([3,5],'a); ? P = x^2-a; polisirreducible(P) %2 = 1 ? [r,m] = ffextend(a, P, 'b); ? r %3 = b^9+2*b^8+b^7+2*b^6+b^4+1 ? subst(ffmap(m, P), x, r) %4 = 0 ? ffgen(r) %5 = b
The library syntax is GEN ffextend(GEN a, GEN P, long v = -1) where v is a variable number.
Return the n-th power of the Frobenius map over the field of definition of m.
? a = ffgen([3,5],'a); ? f = fffrobenius(a); ? ffmap(f,a) == a^3 %3 = 1 ? g = fffrobenius(a, 5); ? ffmap(g,a) == a %5 = 1 ? h = fffrobenius(a, 2); ? h == ffcompomap(f,f) %7 = 1
The library syntax is GEN fffrobenius(GEN m, long n).
Return a generator for the finite field k as a t_FFELT.
The field k can be given by
* its order q
* the pair [p,f] where q = pf
* a monic irreducible polynomial with t_INTMOD coefficients modulo a
prime.
* a t_FFELT belonging to k.
If v is given, the variable name is used to display g, else the
variable of the polynomial or the t_FFELT is used, else x is used.
For efficiency, the characteristic is not checked to be prime; similarly
if a polynomial is given, we do not check whether it is irreducible.
When only the order is specified, the function uses the polynomial generated
by ffinit and is deterministic: two calls to the function with the
same parameters will always give the same generator.
To obtain a multiplicative generator, call ffprimroot on the result
(which is randomized). Its minimal polynomial then gives a primitive
polynomial, which can be used to redefine the finite field so that all
subsequent computations use the new primitive polynomial:
? g = ffgen(16, 't); ? g.mod \\ recover the underlying polynomial. %2 = t^4 + t^3 + t^2 + t + 1 ? g.pol \\ lift g as a t_POL %3 = t ? g.p \\ recover the characteristic %4 = 2 ? fforder(g) \\ g is not a multiplicative generator %5 = 5 ? a = ffprimroot(g) \\ recover a multiplicative generator %6 = t^3 + t^2 + t ? fforder(a) %7 = 15 ? T = minpoly(a) \\ primitive polynomial %8 = Mod(1, 2)*x^4 + Mod(1, 2)*x^3 + Mod(1, 2) ? G = ffgen(T); \\ is now a multiplicative generator ? fforder(G) %10 = 15
The library syntax is GEN ffgen(GEN k, long v = -1) where v is a variable number.
To create a generator for a prime finite field, the function
GEN p_to_GEN(GEN p, long v) returns ffgen(p,v)^0.
Computes a monic polynomial of degree n which is irreducible over 𝔽p, where p is assumed to be prime. This function uses a fast variant of Adleman and Lenstra's algorithm.
It is useful in conjunction with ffgen; for instance if
P = ffinit(3,2), you can represent elements in 𝔽32 in term of
g = ffgen(P,'t). This can be abbreviated as
g = ffgen(3^2, 't), where the defining polynomial P can be later
recovered as g.mod.
The library syntax is GEN ffinit(GEN p, long n, long v = -1) where v is a variable number.
m being a map from K to L two finite fields, return the partial map p from L to K such that for all k ∈ K, p(m(k)) = k.
? a = ffgen([3,5],'a); ? b = ffgen([3,10],'b); ? m = ffembed(a, b); ? p = ffinvmap(m); ? u = random(a); ? v = ffmap(m, u); ? ffmap(p, v^2+v+2) == u^2+u+2 %7 = 1 ? ffmap(p, b) %8 = []
The library syntax is GEN ffinvmap(GEN m).
Discrete logarithm of the finite field element x in base g,
i.e. an e in ℤ such that ge = o. If
present, o represents the multiplicative order of g, see
Section se:DLfun; the preferred format for
this parameter is [ord, factor(ord)], where ord is the
order of g. It may be set as a side effect of calling ffprimroot.
The result is undefined if e does not exist. This function uses
* a combination of generic discrete log algorithms (see znlog)
* a cubic sieve index calculus algorithm for large fields of degree at least 5.
* Coppersmith's algorithm for fields of characteristic at most 5.
? t = ffgen(ffinit(7,5)); ? o = fforder(t) %2 = 5602 \\ not a primitive root. ? fflog(t^10,t) %3 = 10 ? fflog(t^10,t, o) %4 = 10 ? g = ffprimroot(t, &o); ? o \\ order is 16806, bundled with its factorization matrix %6 = [16806, [2, 1; 3, 1; 2801, 1]] ? fforder(g, o) %7 = 16806 ? fflog(g^10000, g, o) %8 = 10000
The library syntax is GEN fflog(GEN x, GEN g, GEN o = NULL).
Given a (partial) map m between two finite fields, return the image of x by m. The function is applied recursively to the component of vectors, matrices and polynomials. If m is a partial map that is not defined at x, return [].
? a = ffgen([3,5],'a); ? b = ffgen([3,10],'b); ? m = ffembed(a, b); ? P = x^2+a*x+1; ? Q = ffmap(m,P); ? ffmap(m,poldisc(P)) == poldisc(Q) %6 = 1
The library syntax is GEN ffmap(GEN m, GEN x).
Given a (partial) map m between two finite fields, express x as an algebraic element over the codomain of m in a way which is compatible with m. The function is applied recursively to the component of vectors, matrices and polynomials.
? a = ffgen([3,5],'a); ? b = ffgen([3,10],'b); ? m = ffembed(a, b); ? mi= ffinvmap(m); ? R = ffmaprel(mi,b) %5 = Mod(b,b^2+(a+1)*b+(a^2+2*a+2))
In particular, this function can be used to compute the relative minimal polynomial, norm and trace:
? minpoly(R) %6 = x^2+(a+1)*x+(a^2+2*a+2) ? trace(R) %7 = 2*a+2 ? norm(R) %8 = a^2+2*a+2
The library syntax is GEN ffmaprel(GEN m, GEN x).
Computes the number of monic irreducible polynomials over 𝔽q of degree exactly n (flag = 0 or omitted) or at most n (flag = 1).
The library syntax is GEN ffnbirred0(GEN q, long n, long flag).
Also available are
GEN ffnbirred(GEN q, long n) (for flag = 0)
and GEN ffsumnbirred(GEN q, long n) (for flag = 1).
Multiplicative order of the finite field element x. If o is
present, it represents a multiple of the order of the element,
see Section se:DLfun; the preferred format for
this parameter is [N, factor(N)], where N is the cardinality
of the multiplicative group of the underlying finite field.
? t = ffgen(ffinit(nextprime(10^8), 5)); ? g = ffprimroot(t, &o); \\ o will be useful! ? fforder(g^1000000, o) time = 0 ms. %5 = 5000001750000245000017150000600250008403 ? fforder(g^1000000) time = 16 ms. \\ noticeably slower, same result of course %6 = 5000001750000245000017150000600250008403
The library syntax is GEN fforder(GEN x, GEN o = NULL).
Return a primitive root of the multiplicative
group of the definition field of the finite field element x (not necessarily
the same as the field generated by x). If present, o is set to
a vector [ord, fa], where ord is the order of the group
and fa its factorization factor(ord). This last parameter is
useful in fflog and fforder, see Section se:DLfun.
? t = ffgen(ffinit(nextprime(10^7), 5)); ? g = ffprimroot(t, &o); ? o[1] %3 = 100000950003610006859006516052476098 ? o[2] %4 = [2 1] [7 2] [31 1] [41 1] [67 1] [1523 1] [10498781 1] [15992881 1] [46858913131 1] ? fflog(g^1000000, g, o) time = 1,312 ms. %5 = 1000000
The library syntax is GEN ffprimroot(GEN x, GEN *o = NULL).
Creates the greatest common divisor of x and y.
If you also need the u and v such that x*u + y*v = gcd(x,y),
use the gcdext function. x and y can have rather quite general
types, for instance both rational numbers. If y is omitted and x is a
vector, returns the gcd of all components of x, i.e. this is
equivalent to content(x).
When x and y are both given and one of them is a vector/matrix type,
the GCD is again taken recursively on each component, but in a different way.
If y is a vector, resp. matrix, then the result has the same type as y,
and components equal to gcd(x, y[i]), resp. gcd(x, y[,i]). Else
if x is a vector/matrix the result has the same type as x and an
analogous definition. Note that for these types, gcd is not
commutative.
The algorithm used is a naive Euclid except for the following inputs:
* integers: use modified right-shift binary ("plus-minus" variant).
* univariate polynomials with coefficients in the same number field (in particular rational): use modular gcd algorithm.
* general polynomials: use the subresultant algorithm if coefficient explosion is likely (non modular coefficients).
If u and v are polynomials in the same variable with inexact coefficients, their gcd is defined to be scalar, so that
? a = x + 0.0; gcd(a,a) %1 = 1 ? b = y*x + O(y); gcd(b,b) %2 = y ? c = 4*x + O(2^3); gcd(c,c) %3 = 4
A good quantitative check to decide whether such a
gcd "should be" nontrivial, is to use polresultant: a value
close to 0 means that a small deformation of the inputs has nontrivial gcd.
You may also use gcdext, which does try to compute an approximate gcd
d and provides u, v to check whether u x + v y is close to d.
The library syntax is GEN ggcd0(GEN x, GEN y = NULL).
Also available are GEN ggcd(GEN x, GEN y), if y is not
NULL, and GEN content(GEN x), if y = NULL.
Returns [u,v,d] such that d is the gcd of x,y, x*u+y*v = gcd(x,y), and u and v minimal in a natural sense. The arguments must be integers or polynomials.
? [u, v, d] = gcdext(32,102) %1 = [16, -5, 2] ? d %2 = 2 ? gcdext(x^2-x, x^2+x-2) %3 = [-1/2, 1/2, x - 1]
If x,y are polynomials in the same variable and inexact
coefficients, then compute u,v,d such that x*u+y*v = d, where d
approximately divides both and x and y; in particular, we do not obtain
gcd(x,y) which is defined to be a scalar in this case:
? a = x + 0.0; gcd(a,a) %1 = 1 ? gcdext(a,a) %2 = [0, 1, x + 0.E-28] ? gcdext(x-Pi, 6*x^2-zeta(2)) %3 = [-6*x - 18.8495559, 1, 57.5726923]
For inexact inputs, the output is thus not well defined mathematically, but you obtain explicit polynomials to check whether the approximation is close enough for your needs.
The library syntax is GEN gcdext0(GEN x, GEN y).
Let inputs x and y be both integers, or both polynomials in the same
variable. Return a vector [M, [a,b]~], where M is an invertible
2 x 2 matrix such that M*[x,y]~ = [a,b]~, where b is
small. More precisely,
* polynomial case: det M has degree 0 and we have deg a ≥ ceil{max(deg x,deg y))/2} > deg b.
* integer case: det M = ± 1 and we have a ≥ ceil{sqrt{max(|x|,|y|)}} > b. Assuming x and y are nonnegative, then M-1 has nonnegative coefficients, and det M is equal to the sign of both main diagonal terms M[1,1] and M[2,2].
The library syntax is GEN ghalfgcd(GEN x, GEN y).
Hilbert symbol of x and y modulo the prime p, p = 0 meaning the place at infinity (the result is undefined if p ! = 0 is not prime).
It is possible to omit p, in which case we take p = 0 if both x
and y are rational, or one of them is a real number. And take p = q
if one of x, y is a t_INTMOD modulo q or a q-adic. (Incompatible
types will raise an error.)
The library syntax is long hilbert(GEN x, GEN y, GEN p = NULL).
True (1) if D is equal to 1 or to the discriminant of a quadratic field, false (0) otherwise. D can be input in factored form as for arithmetic functions:
? isfundamental(factor(-8)) %1 = 1 \\ count fundamental discriminants up to 10^8 ? c = 0; forfactored(d = 1, 10^8, if (isfundamental(d), c++)); c time = 40,840 ms. %2 = 30396325 ? c = 0; for(d = 1, 10^8, if (isfundamental(d), c++)); c time = 1min, 33,593 ms. \\ slower ! %3 = 30396325
The library syntax is long isfundamental(GEN D).
True (1) if the integer x is an s-gonal number, false (0) if not.
The parameter s > 2 must be a t_INT. If N is given, set it to n
if x is the n-th s-gonal number.
? ispolygonal(36, 3, &N) %1 = 1 ? N
The library syntax is long ispolygonal(GEN x, GEN s, GEN *N = NULL).
If k is given, returns true (1) if x is a k-th power, false
(0) if not. What it means to be a k-th power depends on the type of
x; see issquare for details.
If k is omitted, only integers and fractions are allowed for x and the
function returns the maximal k ≥ 2 such that x = nk is a perfect
power, or 0 if no such k exist; in particular ispower(-1),
ispower(0), and ispower(1) all return 0.
If a third argument &n is given and x is indeed a k-th power, sets n to a k-th root of x.
For a t_FFELT x, instead of omitting k (which is
not allowed for this type), it may be natural to set
k = (x.p ^ x.f - 1) / fforder(x)
The library syntax is long ispower(GEN x, GEN k = NULL, GEN *n = NULL).
Also available is
long gisanypower(GEN x, GEN *pty) (k omitted).
True (1) if x is a powerful integer, false (0) if not; an integer is powerful if and only if its valuation at all primes dividing x is greater than 1.
? ispowerful(50) %1 = 0 ? ispowerful(100) %2 = 1 ? ispowerful(5^3*(10^1000+1)^2) %3 = 1
The library syntax is long ispowerful(GEN x).
True (1) if x is a prime number, false (0) otherwise. A prime number is a positive integer having exactly two distinct divisors among the natural numbers, namely 1 and itself.
This routine proves or disproves rigorously that a number is prime, which can
be very slow when x is indeed a large prime integer. For instance
a 1000 digits prime should require 15 to 30 minutes with default algorithms.
Use ispseudoprime to quickly check for compositeness. Use
primecert in order to obtain a primality proof instead of a yes/no
answer; see also factor.
The function accepts vector/matrices arguments, and is then applied componentwise.
If flag = 0, use a combination of
* Baillie-Pomerance-Selfridge-Wagstaff compositeness test
(see ispseudoprime),
* Selfridge "p-1" test if x-1 is smooth enough,
* Adleman-Pomerance-Rumely-Cohen-Lenstra (APRCL) for general medium-sized x (less than 1500 bits),
* Atkin-Morain's Elliptic Curve Primality Prover (ECPP) for general large x.
If flag = 1, use Selfridge-Pocklington-Lehmer "p-1" test; this requires partially factoring various auxilliary integers and is likely to be very slow.
If flag = 2, use APRCL only.
If flag = 3, use ECPP only.
The library syntax is GEN gisprime(GEN x, long flag).
If x = pk is a prime power (p prime, k > 0), return k, else return 0. If a second argument &n is given and x is indeed the k-th power of a prime p, sets n to p.
The library syntax is long isprimepower(GEN x, GEN *n = NULL).
True (1) if x is a strong pseudo
prime (see below), false (0) otherwise. If this function returns false, x
is not prime; if, on the other hand it returns true, it is only highly likely
that x is a prime number. Use isprime (which is of course much
slower) to prove that x is indeed prime.
The function accepts vector/matrices arguments, and is then applied
componentwise.
If flag = 0, checks whether x has no small prime divisors (up to 101 included) and is a Baillie-Pomerance-Selfridge-Wagstaff pseudo prime. Such a pseudo prime passes a Rabin-Miller test for base 2, followed by a Lucas test for the sequence (P,1), where P ≥ 3 is the smallest odd integer such that P2 - 4 is not a square mod x. (Technically, we are using an "almost extra strong Lucas test" that checks whether Vn is ± 2, without computing Un.)
There are no known composite numbers passing the above test, although it is
expected that infinitely many such numbers exist. In particular, all
composites ≤ 264 are correctly detected (checked using
https://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html).
If flag > 0, checks whether x is a strong Miller-Rabin pseudo prime for flag randomly chosen bases (with end-matching to catch square roots of -1).
The library syntax is GEN gispseudoprime(GEN x, long flag).
If x = pk is a pseudo-prime power (p pseudo-prime as per
ispseudoprime, k > 0), return k, else
return 0. If a second argument &n is given and x is indeed
the k-th power of a prime p, sets n to p.
More precisely, k is always the largest integer such that x = nk for
some integer n and, when n ≤ 264 the function returns k > 0 if and
only if n is indeed prime. When n > 264 is larger than the threshold,
the function may return 1 even though n is composite: it only passed
an ispseudoprime(n) test.
The library syntax is long ispseudoprimepower(GEN x, GEN *n = NULL).
True (1) if x is a square, false (0)
if not. What "being a square" means depends on the type of x: all
t_COMPLEX are squares, as well as all nonnegative t_REAL; for
exact types such as t_INT, t_FRAC and t_INTMOD, squares are
numbers of the form s2 with s in ℤ, ℚ and ℤ/Nℤ respectively.
? issquare(3) \\ as an integer %1 = 0 ? issquare(3.) \\ as a real number %2 = 1 ? issquare(Mod(7, 8)) \\ in Z/8Z %3 = 0 ? issquare( 5 + O(13^4) ) \\ in Q_13 %4 = 0
If n is given, a square root of x is put into n.
? issquare(4, &n) %1 = 1 ? n %2 = 2
For polynomials, either we detect that the characteristic is 2 (and check directly odd and even-power monomials) or we assume that 2 is invertible and check whether squaring the truncated power series for the square root yields the original input.
For t_POLMOD x, we only support t_POLMODs of t_INTMODs
encoding finite fields, assuming without checking that the intmod modulus
p is prime and that the polmod modulus is irreducible modulo p.
? issquare(Mod(Mod(2,3), x^2+1), &n) %1 = 1 ? n %2 = Mod(Mod(2, 3)*x, Mod(1, 3)*x^2 + Mod(1, 3))
The library syntax is long issquareall(GEN x, GEN *n = NULL).
Also available is long issquare(GEN x). Deprecated
GP-specific functions GEN gissquare(GEN x) and
GEN gissquareall(GEN x, GEN *pt) return gen0 and gen1
instead of a boolean value.
True (1) if x is squarefree, false (0) if not. Here x can be an integer or a polynomial with coefficients in an integral domain.
? issquarefree(12) %1 = 0 ? issquarefree(6) %2 = 1 ? issquarefree(x^3+x^2) %3 = 0 ? issquarefree(Mod(1,4)*(x^2+x+1)) \\ Z/4Z is not a domain ! *** at top-level: issquarefree(Mod(1,4)*(x^2+x+1)) *** ^ — — — — — — — — — — -- *** issquarefree: impossible inverse in Fp_inv: Mod(2, 4).
A polynomial is declared squarefree if gcd(x,x') is
1. In particular a nonzero polynomial with inexact coefficients is
considered to be squarefree. Note that this may be inconsistent with
factor, which first rounds the input to some exact approximation before
factoring in the apropriate domain; this is correct when the input is not
close to an inseparable polynomial (the resultant of x and x' is not
close to 0).
An integer can be input in factored form as in arithmetic functions.
? issquarefree(factor(6)) %1 = 1 \\ count squarefree integers up to 10^8 ? c = 0; for(d = 1, 10^8, if (issquarefree(d), c++)); c time = 3min, 2,590 ms. %2 = 60792694 ? c = 0; forfactored(d = 1, 10^8, if (issquarefree(d), c++)); c time = 45,348 ms. \\ faster ! %3 = 60792694
The library syntax is long issquarefree(GEN x).
True (1) if x = φ(n) for some integer n, false (0) if not.
? istotient(14) %1 = 0 ? istotient(100) %2 = 0
If N is given, set N = n as well.
? istotient(4, &n) %1 = 1 ? n %2 = 10
The library syntax is long istotient(GEN x, GEN *N = NULL).
Kronecker symbol (x|y), where x and y must be of type integer. By definition, this is the extension of Legendre symbol to ℤ x ℤ by total multiplicativity in both arguments with the following special rules for y = 0, -1 or 2:
* (x|0) = 1 if |x |= 1 and 0 otherwise.
* (x|-1) = 1 if x ≥ 0 and -1 otherwise.
* (x|2) = 0 if x is even and 1 if x = 1,-1 mod 8 and -1 if x = 3,-3 mod 8.
The library syntax is long kronecker(GEN x, GEN y).
Least common multiple of x and y, i.e. such that lcm(x,y)*gcd(x,y) = x*y, up to units. If y is omitted and x is a vector, returns the lcm of all components of x. For integer arguments, return the nonnegative lcm.
When x and y are both given and one of them is a vector/matrix type,
the LCM is again taken recursively on each component, but in a different way.
If y is a vector, resp. matrix, then the result has the same type as y,
and components equal to lcm(x, y[i]), resp. lcm(x, y[,i]). Else
if x is a vector/matrix the result has the same type as x and an
analogous definition. Note that for these types, lcm is not
commutative.
Note that lcm(v) is quite different from
l = v[1]; for (i = 1, #v, l = lcm(l, v[i]))
Indeed, lcm(v) is a scalar, but l may not be (if one of
the v[i] is a vector/matrix). The computation uses a divide-conquer tree
and should be much more efficient, especially when using the GMP
multiprecision kernel (and more subquadratic algorithms become available):
? v = vector(10^5, i, random); ? lcm(v); time = 546 ms. ? l = v[1]; for (i = 1, #v, l = lcm(l, v[i])) time = 4,561 ms.
The library syntax is GEN glcm0(GEN x, GEN y = NULL).
Return the largest non-negative integer e so that be ≤ x, where b > 1 is an integer and x ≥ 1 is a real number. If the parameter z is present, set it to be.
? logint(1000, 2) %1 = 9 ? 2^9 %2 = 512 ? logint(1000, 2, &z) %3 = 9 ? z %4 = 512 ? logint(Pi^2, 2, &z) %5 = 3 ? z %6 = 8
The number of digits used to write x in base b is
1 + logint(x,b):
? #digits(1000!, 10) %5 = 2568 ? logint(1000!, 10) %6 = 2567
This function may conveniently replace
floor( log(x) / log(b) )
which may not give the correct answer since PARI does not guarantee exact rounding.
The library syntax is long logint0(GEN x, GEN b, GEN *z = NULL).
Moebius μ-function of |x|; x must be a nonzero integer.
The library syntax is long moebius(GEN x).
Finds the smallest pseudoprime (see
ispseudoprime) greater than or equal to x. x can be of any real
type. Note that if x is a pseudoprime, this function returns x and not
the smallest pseudoprime strictly larger than x. To rigorously prove that
the result is prime, use isprime.
? nextprime(2) %1 = 2 ? nextprime(Pi) %2 = 5 ? nextprime(-10) %3 = 2 \\ primes are positive
Despite the name, please note that the function is not guaranteed to return
a prime number, although no counter-example is known at present. The return
value is a guaranteed prime if x ≤ 264. To rigorously prove
that the result is prime in all cases, use isprime.
The library syntax is GEN nextprime(GEN x).
Number of divisors of |x|. x must be of type integer.
The library syntax is GEN numdiv(GEN x).
Number of distinct prime divisors of |x|. x must be of type integer.
? factor(392) %1 = [2 3] [7 2] ? omega(392) %2 = 2; \\ without multiplicity ? bigomega(392) %3 = 5; \\ = 3+2, with multiplicity
The library syntax is long omega(GEN x).
Finds the largest pseudoprime (see ispseudoprime) less than or equal
to x; the input x can be of any real type.
Returns 0 if x ≤ 1. Note that if x is a prime, this function returns x
and not the largest prime strictly smaller than x.
? precprime(2) %1 = 2 ? precprime(Pi) %2 = 3 ? precprime(-10) %3 = 0 \\ primes are positive
The function name comes from preceding prime.
Despite the name, please note that the function is not guaranteed to return
a prime number (although no counter-example is known at present); the return
value is a guaranteed prime if x ≤ 264. To rigorously prove
that the result is prime in all cases, use isprime.
The library syntax is GEN precprime(GEN x).
The n-th prime number
? prime(10^9) %1 = 22801763489
Uses checkpointing and a naive O(n) algorithm. Will need
about 30 minutes for n up to 1011; make sure to start gp with
primelimit at least sqrt{pn}, e.g. the value
sqrt{nlog (nlog n)} is guaranteed to be sufficient.
The library syntax is GEN prime(long n).
If N is a prime, return a PARI Primality Certificate for the prime N,
as described below. Otherwise, return 0. A Primality Certificate
c can be checked using primecertisvalid(c).
If flag = 0 (default), return an ECPP certificate (Atkin-Morain)
If flag = 0 and partial > 0, return a (potentially) partial ECPP certificate.
A PARI ECPP Primality Certificate for the prime N is either a prime
integer N < 264 or a vector C of length ℓ whose ith
component C[i] is a vector [Ni, ti, si, ai, Pi]
of length 5
where N1 = N. It is said to be valid if for each
i = 1,..., ℓ, all of the following conditions are satisfied
* Ni is a positive integer
* ti is an integer such that ti2 < 4Ni
* si is a positive integer which divides mi where mi = Ni + 1 - ti
* If we set qi = (mi)/(si), then
* qi > (Ni1/4+1)2
* qi = Ni+1 if 1 ≤ i < l
* qℓ ≤ 264 is prime
* ai is an integer
* P[i] is a vector of length 2 representing the affine
point Pi = (xi, yi) on the elliptic curve
E: y2 = x3 + aix + bi modulo Ni where
bi = yi2 - xi3 - aixi satisfying the following:
* mi Pi = oo
* si Pi ! = oo
Using the following theorem, the data in the vector C allows to
recursively certify the primality of N (and all the qi) under the single
assumption that qℓ be prime.
Theorem. If N is an integer and there exist positive integers m, q and a point P on the elliptic curve E: y2 = x3 + ax + b defined modulo N such that q > (N1/4 + 1)2, q is a prime divisor of m, mP = oo and (m)/(q)P ! = oo , then N is prime.
A partial certificate is identical except that the condition qℓ ≤ 264 is replaced by qℓ ≤ 2partial. Such partial certificate C can be extended to a full certificate by calling C = primecert(C), or to a longer partial certificate by calling C = primecert(C,,b) with b < partial.
? primecert(10^35 + 69) %1 = [[100000000000000000000000000000000069, 5468679110354 52074, 2963504668391148, 0, [60737979324046450274283740674 208692, 24368673584839493121227731392450025]], [3374383076 4501150277, -11610830419, 734208843, 0, [26740412374402652 72 4, 6367191119818901665]], [45959444779, 299597, 2331, 0 , [18022351516, 9326882 51]]] ? primecert(nextprime(2^64)) %2 = [[18446744073709551629, -8423788454, 160388, 1, [1059 8342506117936052, 2225259013356795550]]] ? primecert(6) %3 = 0 ? primecert(41) %4 = 41 ? N = 2^2000+841; ? Cp1 = primecert(N,,1500); \\ partial certificate time = 16,018 ms. ? Cp2 = primecert(Cp1,,1000); \\ (longer) partial certificate time = 5,890 ms. ? C = primecert(Cp2); \\ full certificate for N time = 1,777 ms. ? primecertisvalid(C) %9 = 1 ? primecert(N); time = 23,625 ms.
As the last command shows, attempting a succession of partial certificates should be about as fast as a direct computation.
If flag = 1 (very slow), return an N-1 certificate (Pocklington Lehmer)
A PARI N-1 Primality Certificate for the prime N is either a prime integer N < 264 or a pair [N, C], where C is a vector with ℓ elements which are either a single integer pi < 264 or a triple [pi,ai,Ci] with pi > 264 satisfying the following properties:
* pi is a prime divisor of N - 1;
* ai is an integer such that aiN-1 = 1 (mod N) and ai(N-1)/pi - 1 is coprime with N;
* Ci is an N-1 Primality Certificate for pi
* The product F of the pivp_{i(N-1)} is strictly larger than N1/3. Provided that all pi are indeed primes, this implies that any divisor of N is congruent to 1 modulo F.
* The Brillhart-Lehmer-Selfridge criterion is satisfied: when we write N = 1 + c1 F + c2 F2 in base F the polynomial 1 + c1 X + c2 X2 is irreducible over ℤ, i.e. c12 - 4c2 is not a square. This implies that N is prime.
This algorithm requires factoring partially p-1 for various prime integers p with an unfactored parted ≤ p2/3 and this may be exceedingly slow compared to the default.
The algorithm fails if one of the pseudo-prime factors is not prime, which is
exceedingly unlikely and well worth a bug report. Note that if you monitor
the algorithm at a high enough debug level, you may see warnings about
untested integers being declared primes. This is normal: we ask for partial
factorizations (sufficient to prove primality if the unfactored part is not
too large), and factor warns us that the cofactor hasn't been tested.
It may or may not be tested later, and may or may not be prime. This does
not affect the validity of the whole Primality Certificate.
The library syntax is GEN primecert0(GEN N, long flag, long partial).
Also available is
GEN ecpp0(GEN N, long partial) (flag = 0).
Returns a string suitable for print/write to display a primality certificate
from primecert, the format of which depends on the value of format:
* 0 (default): Human-readable format. See ??primecert for the
meaning of the successive N, t, s, a, m, q, E, P. The integer D is the
negative fundamental discriminant coredisc(t2 - 4N).
* 1: Primo format 4.
* 2: MAGMA format.
Currently, only ECPP Primality Certificates are supported.
? cert = primecert(10^35+69);
? s = primecertexport(cert); \\ Human-readable
? print(s)
[1]
N = 100000000000000000000000000000000069
t = 546867911035452074
s = 2963504668391148
a = 0
D = -3
m = 99999999999999999453132088964547996
q = 33743830764501150277
E = [0, 1]
P = [21567861682493263464353543707814204,
49167839501923147849639425291163552]
[2]
N = 33743830764501150277
t = -11610830419
s = 734208843
a = 0
D = -3
m = 33743830776111980697
q = 45959444779
E = [0, 25895956964997806805]
P = [29257172487394218479, 3678591960085668324]
\\ Primo format
? s = primecertexport(cert,1); write("cert.out", s);
\\ Magma format, write to file
? s = primecertexport(cert,2); write("cert.m", s);
? cert = primecert(10^35+69, 1); \\ N-1 certificate
? primecertexport(cert)
*** at top-level: primecertexport(cert)
*** ^ — — — — — — —
*** primecertexport: sorry, N-1 certificate is not yet implemented.
The library syntax is GEN primecertexport(GEN cert, long format).
Verifies if cert is a valid PARI ECPP Primality certificate, as described
in ??primecert.
? cert = primecert(10^35 + 69) %1 = [[100000000000000000000000000000000069, 5468679110354 52074, 2963504668391148, 0, [60737979324046450274283740674 208692, 24368673584839493121227731392450025]], [3374383076 4501150277, -11610830419, 734208843, 0, [26740412374402652 72 4, 6367191119818901665]], [45959444779, 299597, 2331, 0 , [18022351516, 9326882 51]]] ? primecertisvalid(cert) %2 = 1 ? cert[1][1]++; \\ random perturbation ? primecertisvalid(cert) %4 = 0 \\ no longer valid ? primecertisvalid(primecert(6)) %5 = 0
The library syntax is long primecertisvalid(GEN cert).
The prime counting function. Returns the number of primes p, p ≤ x.
? primepi(10) %1 = 4; ? primes(5) %2 = [2, 3, 5, 7, 11] ? primepi(10^11) %3 = 4118054813
Uses checkpointing and a naive O(x) algorithm;
make sure to start gp with primelimit at least sqrt{x}.
The library syntax is GEN primepi(GEN x).
Creates a row vector whose components are the first n prime numbers.
(Returns the empty vector for n ≤ 0.) A t_VEC n = [a,b] is also
allowed, in which case the primes in [a,b] are returned
? primes(10) \\ the first 10 primes %1 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] ? primes([0,29]) \\ the primes up to 29 %2 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] ? primes([15,30]) %3 = [17, 19, 23, 29]
The library syntax is GEN primes0(GEN n).
Ordinary class number of the quadratic order of discriminant D, for "small" values of D.
* if D > 0 or flag = 1, use a O(|D|1/2)
algorithm (compute L(1,χD) with the approximate functional equation).
This is slower than quadclassunit as soon as |D| ~ 102 or
so and is not meant to be used for large D.
* if D < 0 and flag = 0 (or omitted), use a O(|D|1/4)
algorithm (Shanks's baby-step/giant-step method). It should
be faster than quadclassunit for small values of D, say
|D| < 1018.
Important warning. In the latter case, this function only
implements part of Shanks's method (which allows to speed it up
considerably). It gives unconditionnally correct results for
|D| < 2.1010, but may give incorrect results for larger values
if the class
group has many cyclic factors. We thus recommend to double-check results
using the function quadclassunit, which is about 2 to 3 times slower in
the range |D| ∈ [1010, 1018], assuming GRH. We currently have no
counter-examples but they should exist: we would appreciate a bug report if
you find one.
Warning. Contrary to what its name implies, this routine does not compute the number of classes of binary primitive forms of discriminant D, which is equal to the narrow class number. The two notions are the same when D < 0 or the fundamental unit ϵ has negative norm; when D > 0 and Nϵ > 0, the number of classes of forms is twice the ordinary class number. This is a problem which we cannot fix for backward compatibility reasons. Use the following routine if you are only interested in the number of classes of forms:
? QFBclassno(D) = qfbclassno(D) * if (D > 0 && quadunitnorm(D) > 0, 2, 1) ? QFBclassno(136) %1 = 4 ? qfbclassno(136) %2 = 2 ? quadunitnorm(136) %3 = 1 ? bnfnarrow(bnfinit(x^2 - 136)).cyc %4 = [4] \\ narrow class group is cyclic ~ Z/4Z
Note that the use of bnfnarrow above is only valid
because 136 is a fundamental discriminant: that function is asymptotically
faster (and returns the group structure, not only its order) but only supports
maximal orders.
Here are a few more examples:
? qfbclassno(400000028) \\ D > 0: slow
time = 3,140 ms.
%1 = 1
? quadclassunit(400000028).no
time = 20 ms. \\ { much faster, assume GRH}
%2 = 1
? qfbclassno(-400000028) \\ D < 0: fast enough
time = 0 ms.
%3 = 7253
? quadclassunit(-400000028).no
time = 0 ms.
%4 = 7253
See also qfbhclassno.
The library syntax is GEN qfbclassno0(GEN D, long flag).
composition of the binary quadratic forms x and y, with reduction of the result.
? x=Qfb(2,3,-10);y=Qfb(5,3,-4); ? qfbcomp(x,y) %2 = Qfb(-2, 9, 1) ? qfbcomp(x,y)==qfbred(qfbcompraw(x,y)) %3 = 1
The library syntax is GEN qfbcomp(GEN x, GEN y).
composition of the binary quadratic forms x and y, without reduction of the result. This is useful e.g. to compute a generating element of an ideal. The result is undefined if x and y do not have the same discriminant.
? x=Qfb(2,3,-10);y=Qfb(5,3,-4);
? qfbcompraw(x,y)
%2 = Qfb(10, 3, -2)
? x=Qfb(2,3,-10);y=Qfb(1,-1,1);
? qfbcompraw(x,y)
*** at top-level: qfbcompraw(x,y)
*** ^ — — — — —
*** qfbcompraw: inconsistent qfbcompraw t_QFB , t_QFB.
The library syntax is GEN qfbcompraw(GEN x, GEN y).
Solves the equation x2 + dy2 = n in integers x and y, where
d > 0 and n is prime. Returns the empty vector [] when no solution
exists. It is also allowed to try n = 4 times a prime but the answer is
then guaranteed only if d is 3 mod 4; more precisely if d ! = 3 mod
4, the algorithm may fail to find a non-primitive solution.
This function is a special case of qfbsolve applied to the principal
form in the imaginary quadratic order of discriminant -4d (returning the
solution with non-negative x and y). As its name implies,
qfbcornacchia uses Cornacchia's algorithm and runs in time quasi-linear
in log n (using halfgcd); in practical ranges, qfbcornacchia
should be about twice faster than qfbsolve unless we indicate to the
latter that its second argument is prime (see below).
? qfbcornacchia(1, 113) %1 = [8, 7] ? qfbsolve(Qfb(1,0,1), 113) %2 = [8, 7] ? qfbcornacchia(1, 4*113) \\ misses the non-primitive solution 2*[8,7] %3 = [] ? qfbcornacchia(1, 4*109) \\ finds a non-primitive solution %4 = [20, 6] ? p = 122838793181521; isprime(p) %5 = 1 ? qfbcornacchia(24, p) %6 = [10547339, 694995] ? Q = Qfb(1,0,24); qfbsolve(Q,p) %7 = [10547339, 694995] ? for (i=1, 10^5, qfbsolve(Q, p)) time = 345 ms. ? for (i=1, 10^5, qfbcornacchia(24,p)) \\ faster time = 251 ms. ? for (i=1, 10^5, qfbsolve(Q, Mat([p,1]))) \\ just as fast time = 251 ms.
We used Mat([p,1]) to indicate that p1
was the integer factorization of p, i.e., that p is prime. Without it,
qfbsolve attempts to factor p and wastes a little time.
The library syntax is GEN qfbcornacchia(GEN d, GEN n).
Hurwitz class number of x, when
x is nonnegative and congruent to 0 or 3 modulo 4, and 0 for other
values. For x > 5.105, we assume the GRH, and use quadclassunit
with default parameters.
? qfbhclassno(1) \\ not 0 or 3 mod 4 %1 = 0 ? qfbhclassno(3) %2 = 1/3 ? qfbhclassno(4) %3 = 1/2 ? qfbhclassno(23) %4 = 3
The library syntax is GEN hclassno(GEN x).
composition of the primitive positive
definite binary quadratic forms x and y (type t_QFB) using the NUCOMP
and NUDUPL algorithms of Shanks, à la Atkin. L is any positive
constant, but for optimal speed, one should take L = |D/4|1/4, i.e.
sqrtnint(abs(D) >> 2,4), where D is the common discriminant of x and
y. When x and y do not have the same discriminant, the result is
undefined.
The current implementation is slower than the generic routine for small D, and becomes faster when D has about 45 bits.
The library syntax is GEN nucomp(GEN x, GEN y, GEN L).
Also available is GEN nudupl(GEN x, GEN L) when x = y.
n-th power of the primitive positive definite
binary quadratic form x using Shanks's NUCOMP and NUDUPL algorithms;
if set, L should be equal to sqrtnint(abs(D) >> 2,4), where D < 0 is
the discriminant of x.
The current implementation is slower than the generic routine for small discriminant D, and becomes faster for D ~ 245.
The library syntax is GEN nupow(GEN x, GEN n, GEN L = NULL).
n-th power of the binary quadratic form
x, computed with reduction (i.e. using qfbcomp).
The library syntax is GEN qfbpow(GEN x, GEN n).
n-th power of the binary quadratic form
x, computed without doing any reduction (i.e. using qfbcompraw).
Here n must be nonnegative and n < 231.
The library syntax is GEN qfbpowraw(GEN x, long n).
Prime binary quadratic form of discriminant
x whose first coefficient is p, where |p| is a prime number.
By abuse of notation,
p = ± 1 is also valid and returns the unit form. Returns an
error if x is not a quadratic residue mod p, or if x < 0 and p < 0.
(Negative definite t_QFB are not implemented.)
The library syntax is GEN primeform(GEN x, GEN p).
Reduces the binary quadratic form x (updating Shanks's distance function d if x = [q,d] is an extended indefinite form). If flag is 1, the function performs a single reduction step, and a complete reduction otherwise.
The arguments isd, sd, if present, supply the values of floor{sqrt{D}}, and sqrt{D} respectively, where D is the discriminant (this is not checked). If d < 0 these values are useless.
The library syntax is GEN qfbred0(GEN x, long flag, GEN isd = NULL, GEN sd = NULL).
Also available is GEN qfbred(GEN x) (flag is 0, isd
and sd are NULL)
Reduction of the (real or imaginary) binary quadratic form x, returns
[y,g] where y is reduced and g in SL(2,ℤ) is such that
g.x = y; isD, if
present, must be equal to sqrtint(D), where D > 0 is the
discriminant of x.
The action of g on x can be computed using qfeval(x,g)
? q1 = Qfb(33947,-39899,11650); ? [q2,U] = qfbredsl2(q1) %2 = [Qfb(749,2207,-1712),[-1,3;-2,5]] ? qfeval(q1,U) %3 = Qfb(749,2207,-1712)
The library syntax is GEN qfbredsl2(GEN x, GEN isD = NULL).
Solve the equation Q(x,y) = n in coprime integers x and y (primitive solutions), where Q is a binary quadratic form and n an integer, up to the action of the special orthogonal group G = SO(Q,ℤ), which is isomorphic to the group of units of positive norm of the quadratic order of discriminant D = disc Q. If D > 0, G is infinite. If D < -4, G is of order 2, if D = -3, G is of order 6 and if D = -4, G is of order 4.
Binary digits of flag mean: 1: return all solutions if set, else a single solution; return [] if a single solution is wanted (bit unset) but none exist. 2: also include imprimitive solutions.
When flag = 2 (return a single solution, possibly imprimitive), the algorithm returns a solution with minimal content; in particular, a primitive solution exists if and only if one is returned.
The integer n can also be given by its factorization matrix
fa = factor(n) or by the pair [n, fa].
? qfbsolve(Qfb(1,0,2), 603) \\ a single primitive solution
%1 = [5, 17]
? qfbsolve(Qfb(1,0,2), 603, 1) \\ all primitive solutions
%2 = [[5, 17], [-19, -11], [19, -11], [5, -17]]
? qfbsolve(Qfb(1,0,2), 603, 2) \\ a single, possibly imprimitive solution
%3 = [5, 17] \\ actually primitive
? qfbsolve(Qfb(1,0,2), 603, 3) \\ all solutions
%4 = [[5, 17], [-19, -11], [19, -11], [5, -17], [-21, 9], [-21, -9]]
? N = 2^128+1; F = factor(N);
? qfbsolve(Qfb(1,0,1),[N,F],1)
%3 = [[-16382350221535464479,8479443857936402504],
[18446744073709551616,-1],[-18446744073709551616,-1],
[16382350221535464479,8479443857936402504]]
For fixed Q, assuming the factorisation of n is given, the algorithm
runs in probabilistic polynomial time in log p, where p is the largest
prime divisor of n, through the computation of square roots of D modulo
4 p). The dependency on Q is more complicated: polynomial time in log
|D| if Q is imaginary, but exponential time if Q is real (through the
computation of a full cycle of reduced forms). In the latter case, note that
bnfisprincipal provides a solution in heuristic subexponential time
assuming the GRH.
The library syntax is GEN qfbsolve(GEN Q, GEN n, long flag).
Buchmann-McCurley's sub-exponential algorithm for computing the class group of a quadratic order of discriminant D. By default, the results are conditional on the GRH.
This function should be used instead of qfbclassno or
quadregulator
when D < -1025, D > 1010, or when the structure is wanted. It
is a special case of bnfinit, which is slower, but more robust.
The result is a vector v whose components should be accessed using member functions:
* v.no: the class number
* v.cyc: a vector giving the structure of the class group as a
product of cyclic groups;
* v.gen: a vector giving generators of those cyclic groups (as
binary quadratic forms).
* v.reg: the regulator, computed to an accuracy which is the
maximum of an internal accuracy determined by the program and the current
default (note that once the regulator is known to a small accuracy it is
trivial to compute it to very high accuracy, see the tutorial).
* v.normfu (for positive D only) return the norm of the
fundamental unit, either 1 or -1. Note that a result of -1 is
unconditional and no longer depends on the GRH.
The flag is obsolete and should be left alone. In older versions,
it supposedly computed the narrow class group when D > 0, but this did not
work at all; use the general function bnfnarrow.
Optional parameter tech is a row vector of the form [c1, c2], where c1 ≤ c2 are nonnegative real numbers which control the execution time and the stack size, see se:GRHbnf. The parameter is used as a threshold to balance the relation finding phase against the final linear algebra. Increasing the default c1 means that relations are easier to find, but more relations are needed and the linear algebra will be harder. The default value for c1 is 0 and means that it is taken equal to c2. The parameter c2 is mostly obsolete and should not be changed, but we still document it for completeness: we compute a tentative class group by generators and relations using a factorbase of prime ideals ≤ c1 (log |D|)2, then prove that ideals of norm ≤ c2 (log |D|)2 do not generate a larger group. By default an optimal c2 is chosen, so that the result is provably correct under the GRH — a result of Grenié and Molteni states that c2 = 23/6 ~ 3.83 is fine (and even c2 = 15/4 ~ 3.75 for large |D| > 2.41 E8). But it is possible to improve on this algorithmically. You may provide a smaller c2, it will be ignored (we use the provably correct one); you may provide a larger c2 than the default value, which results in longer computing times for equally correct outputs (under GRH).
The library syntax is GEN quadclassunit0(GEN D, long flag, GEN tech = NULL, long prec).
If you really need to experiment with the tech parameter,
it will be more convenient to use
GEN Buchquad(GEN D, double c1, double c2, long prec).
Discriminant of the étale algebra ℚ(sqrt{x}), where x ∈ ℚ*.
This is the same as coredisc(d) where d is the integer
squarefree part of x, so x = d f2 with f ∈ ℚ* and d ∈ ℤ.
This returns 0 for x = 0, 1 for x square and the discriminant of
the quadratic field ℚ(sqrt{x}) otherwise.
? quaddisc(7) %1 = 28 ? quaddisc(-7) %2 = -7
The library syntax is GEN quaddisc(GEN x).
Creates the quadratic number ω = (a+sqrt{D})/2 where a = 0 if D = 0 mod 4, a = 1 if D = 1 mod 4, so that (1,ω) is an integral basis for the quadratic order of discriminant D. D must be an integer congruent to 0 or 1 modulo 4, which is not a square. If v is given, the variable name is used to display g else 'w' is used.
? w = quadgen(5, 'w); w^2 - w - 1 %1 = 0 ? w = quadgen(0, 'w) *** at top-level: w=quadgen(0) *** ^ — — — - *** quadgen: domain error in quadpoly: issquare(disc) = 1
The library syntax is GEN quadgen0(GEN D, long v = -1) where v is a variable number.
When v does not matter, the function
GEN quadgen(GEN D) is also available.
Relative equation defining the Hilbert class field of the quadratic field of discriminant D.
If D < 0, uses complex multiplication (Schertz's variant).
If D > 0 Stark units are used and (in rare cases) a
vector of extensions may be returned whose compositum is the requested class
field. See bnrstark for details.
The library syntax is GEN quadhilbert(GEN D, long prec).
Creates the "canonical" quadratic
polynomial (in the variable v) corresponding to the discriminant D,
i.e. the minimal polynomial of quadgen(D). D must be an integer
congruent to 0 or 1 modulo 4, which is not a square.
? quadpoly(5,'y) %1 = y^2 - y - 1 ? quadpoly(0,'y) *** at top-level: quadpoly(0,'y) *** ^ — — — — -- *** quadpoly: domain error in quadpoly: issquare(disc) = 1
The library syntax is GEN quadpoly0(GEN D, long v = -1) where v is a variable number.
Relative equation for the ray
class field of conductor f for the quadratic field of discriminant D
using analytic methods. A bnf for x2 - D is also accepted in place
of D.
For D < 0, uses the σ function and Schertz's method.
For D > 0, uses Stark's conjecture, and a vector of relative equations may be
returned. See bnrstark for more details.
The library syntax is GEN quadray(GEN D, GEN f, long prec).
Regulator of the quadratic order of positive discriminant D in time
Õ(D1/2) using the continued fraction algorithm. Raise
an error if D is not a discriminant (fundamental or not) or if D is a
square. The function quadclassunit is asymptotically faster (and also
in practice for D > 1010 or so) but depends on the GRH.
The library syntax is GEN quadregulator(GEN D, long prec).
A fundamental unit u of the real quadratic order
of discriminant D. The integer D must be congruent to 0 or 1 modulo 4
and not a square; the result is a quadratic number (see Section se:quadgen).
If D is not a fundamental discriminant, the algorithm is wasteful: if D =
df2 with d fundamental, it will be faster to compute quadunit(d)
then raise it to the power quadunitindex(d,f); or keep it in
factored form.
If v is given, the variable name is used to display u
else 'w' is used. The algorithm computes the continued fraction
of (1 + sqrt{D}) / 2 or sqrt{D}/2 (see GTM 138, algorithm 5.7.2).
Although the continued fraction length is only O(sqrt{D}),
the function still runs in time Õ(D), in part because the
output size is not polynomially bounded in terms of log D.
See bnfinit and bnfunits for a better alternative for large
D, running in time subexponential in log D and returning the
fundamental units in compact form (as a short list of S-units of size
O(log D)3 raised to possibly large exponents).
The library syntax is GEN quadunit0(GEN D, long v = -1) where v is a variable number.
When v does not matter, the function
GEN quadunit(GEN D) is also available.
Given a fundamental discriminant D, returns the index of the unit group
of the order of conductor f in the units of ℚ(sqrt{D}). This function
uses the continued fraction algorithm and has O(D1/2 + ϵ
fϵ) complexity; quadclassunit is asymptotically faster but
depends on the GRH.
? quadunitindex(-3, 2) %1 = 3 ? quadunitindex(5, 2^32) \\ instantaneous %2 = 3221225472 ? quadregulator(5 * 2^64) / quadregulator(5) time = 3min, 1,488 ms. %3 = 3221225472.0000000000000000000000000000
The conductor f can be given in factored form or as
[f, factor(f)]:
? quadunitindex(5, [100, [2,2;5,2]]) %4 = 150 ? quadunitindex(5, 100) %5 = 150 ? quadunitindex(5, [2,2;5,2]) %6 = 150
If D is not fundamental, the result is undefined; you may use the following script instead:
index(d, f) =
{ my([D,F] = coredisc(d, 1));
quadunitindex(D, f * F) / quadunitindex(D, F)
}
? index(5 * 10^2, 10)
%7 = 10
The library syntax is GEN quadunitindex(GEN D, GEN f).
Returns the norm (1 or -1) of the fundamental unit of the quadratic
order of discriminant D. The integer D must be congruent to 0 or 1
modulo 4 and not a square. This is of course equal to norm(quadunit(D))
but faster.
? quadunitnorm(-3) \\ the result is always 1 in the imaginary case %1 = 1 ? quadunitnorm(5) %2 = -1 ? quadunitnorm(17345) %3 = -1 ? u = quadunit(17345) %4 = 299685042291 + 4585831442*w ? norm(u) %5 = -1
This function computes the parity of the continued fraction
expansion and runs in time Õ(D1/2). If D is fundamental,
the function bnfinit is asymptotically faster but depends of the GRH.
If D = df2 is not fundamental, it will usually be faster to first compute
quadunitindex(d, f). If it is even, the result is 1, else the result
is quadunitnorm(d). The narrow class number of the order of
discriminant D is equal to the class number if the unit norm is 1 and to
twice the class number otherwise.
Important remark. Assuming GRH, using bnfinit is much
faster, running in time subexponential in log D (instead of exponential
for quadunitnorm). We give examples for the maximal order:
? GRHunitnorm(bnf) = vecprod(bnfsignunit(bnf)[,1]) ? bnf = bnfinit(x^2 - 17345, 1); GRHunitnorm(bnf) %2 = -1 ? bnf = bnfinit(x^2 - nextprime(2^60), 1); GRHunitnorm(bnf) time = 119 ms. %3 = -1 ? quadunitnorm(nextprime(2^60)) time = 24,086 ms. %4 = -1
Note that if the result is -1, it is unconditional because (if GRH is false) it could happen that our tentative fundamental unit in bnf is actually a power uk of the true fundamental unit, but we would still have Norm(u) = -1 (and k odd). We can also remove the GRH assumption when the result is 1 with a little more work:
? v = bnfunits(bnf)[1][1] \\ a unit in factored form ? v[,2] %= 2; ? nfeltissquare(bnf, nffactorback(bnf, v)) %7 = 0
Under GRH, we know that v is the fundamental unit, but as
above it can be a power uk of the true fundamental unit u. But the
final two lines prove that v is not a square, hence k is odd and
Norm(u) must also be 1. We modified the factorization matrix
giving v by reducing all exponents modulo 2: this allows to computed
nffactorback even when the factorization involves huge exponents.
And of course the new v is a square if and only if the original one was.
The library syntax is long quadunitnorm(GEN D).
Compute the value of Ramanujan's tau function at an individual n,
assuming the truth of the GRH (to compute quickly class numbers of imaginary
quadratic fields using quadclassunit). If ell is 16, 18, 20, 22,
or 26, same for the newform of level 1 and corresponding weight. Otherwise,
compute the coefficient of the trace form at n.
The complexity is in Õ(n1/2) using O(log n) space.
If all values up to N are required, then
∑ τ(n)qn = q ∏n ≥ 1 (1-qn)24
and more generally, setting u = ℓ - 13 and C = 2/ζ(-u) for ℓ
> 12,
∑τℓ(n)qn = q ∏n ≥ 1
(1-qn)24 ( 1 + C∑n ≥ 1nu qn / (1-qn))
produces them in time Õ(N), against Õ(N3/2) for
individual calls to ramanujantau; of course the space complexity then
becomes Õ(N). For other values of ℓ,
mfcoefs(mftraceform([1,ell]),N) is much faster.
? tauvec(N) = Vec(q*eta(q + O(q^N))^24); ? N = 10^4; v = tauvec(N); time = 26 ms. ? ramanujantau(N) %3 = -482606811957501440000 ? w = vector(N, n, ramanujantau(n)); \\ much slower ! time = 13,190 ms. ? v == w %4 = 1
The library syntax is GEN ramanujantau(GEN n, long ell).
Returns a strong pseudo prime (see ispseudoprime) in [2,N-1].
A t_VEC N = [a,b] is also allowed, with a ≤ b in which case a
pseudo prime a ≤ p ≤ b is returned; if no prime exists in the
interval, the function will run into an infinite loop. If the upper bound
is less than 264 the pseudo prime returned is a proven prime.
? randomprime(100) %1 = 71 ? randomprime([3,100]) %2 = 61 ? randomprime([1,1]) *** at top-level: randomprime([1,1]) *** ^ — — — — — — *** randomprime: domain error in randomprime: *** floor(b) - max(ceil(a),2) < 0 ? randomprime([24,28]) \\ infinite loop
If the optional parameter q is an integer, return a prime congruent to 1 mod q; if q is an intmod, return a prime in the given congruence class. If the class contains no prime in the given interval, the function will raise an exception if the class is not invertible, else run into an infinite loop
? randomprime(100, 4) \\ 1 mod 4
%1 = 71
? randomprime(100, 4)
%2 = 13
? randomprime([10,100], Mod(2,5))
%3 = 47
? randomprime(100, Mod(0,2)) \\ silly but works
%4 = 2
? randomprime([3,100], Mod(0,2)) \\ not invertible
*** at top-level: randomprime([3,100],Mod(0,2))
*** ^ — — — — — — — — — --
*** randomprime: elements not coprime in randomprime:
0
2
? randomprime(100, 97) \\ infinite loop
The library syntax is GEN randomprime0(GEN N = NULL, GEN q = NULL).
Also available is GEN randomprime(GEN N = NULL).
Removes the primes listed in x from
the prime number table. In particular removeprimes(addprimes()) empties
the extra prime table. x can also be a single integer. List the current
extra primes if x is omitted.
The library syntax is GEN removeprimes(GEN x = NULL).
Sum of the k-th powers of the positive divisors of |x|. x and k must be of type integer.
The library syntax is GEN sumdivk(GEN x, long k).
Also available is GEN sumdiv(GEN n), for k = 1.
Returns the integer square root of x, i.e. the largest integer y such that y2 ≤ x, where x a nonnegative real number. If r is present, set it to the remainder r = x - y2, which satisfies 0 ≤ r < 2y + 1. Further, when x is an integer, r is an integer satisfying 0 ≤ r ≤ 2y.
? x = 120938191237; sqrtint(x) %1 = 347761 ? sqrt(x) %2 = 347761.68741970412747602130964414095216 ? y = sqrtint(x, &r); r %3 = 478116 ? x - y^2 %4 = 478116 ? sqrtint(9/4, &r) \\ not 3/2 ! %5 = 1 ? r %6 = 5/4
The library syntax is GEN sqrtint0(GEN x, GEN *r = NULL).
Also available is GEN sqrtint(GEN a).
Returns the integer n-th root of x, i.e. the largest integer y such that yn ≤ x, where x is a nonnegative real number.
? N = 120938191237; sqrtnint(N, 5) %1 = 164 ? N^(1/5) %2 = 164.63140849829660842958614676939677391 ? sqrtnint(Pi^2, 3) %3 = 2
The special case n = 2 is sqrtint
The library syntax is GEN sqrtnint(GEN x, long n).
Returns the Dedekind sum attached to the integers h and k, corresponding to a fast implementation of
s(h,k) = sum(n = 1, k-1, (n/k)*(frac(h*n/k) - 1/2))
The library syntax is GEN sumdedekind(GEN h, GEN k).
Sum of digits in the integer n, when written in base B.
? sumdigits(123456789) %1 = 45 ? sumdigits(123456789, 2) %2 = 16 ? sumdigits(123456789, -2) %3 = 15
Note that the sum of bits in n is also returned by
hammingweight. This function is much faster than
vecsum(digits(n,B)) when B is 10 or a power of 2, and only
slightly faster in other cases.
The library syntax is GEN sumdigits0(GEN n, GEN B = NULL).
Also available is GEN sumdigits(GEN n), for B = 10.
Given a datum D describing a group (ℤ/Nℤ)* and a Dirichlet
character χ, return the pair [G, chi], where G is
znstar(N, 1)) and chi is a GP character.
The following possibilities for D are supported
* a nonzero t_INT congruent to 0,1 modulo 4, return the real
character modulo D given by the Kronecker symbol (D/.);
* a t_INTMOD Mod(m, N), return the Conrey character
modulo N of index m (see znconreylog).
* a modular form space as per mfinit([N,k,χ]) or a modular
form for such a space, return the underlying Dirichlet character χ
(which may be defined modulo a divisor of N but need not be primitive).
In the remaining cases, G is initialized by znstar(N, 1).
* a pair [G, chi], where chi is a standard GP Dirichlet
character c = (cj) on G (generic character t_VEC or
Conrey characters t_COL or t_INT); given
generators G = ⨁ (ℤ/djℤ) gj, χ(gj) = e(cj/dj).
* a pair [G, chin], where chin is a normalized
representation [n, ~{c}] of the Dirichlet character c; χ(gj)
= e(~{c}j / n) where n is minimal (order of χ).
? [G,chi] = znchar(-3); ? G.cyc %2 = [2] ? chareval(G, chi, 2) %3 = 1/2 ? kronecker(-3,2) %4 = -1 ? znchartokronecker(G,chi) %5 = -3 ? mf = mfinit([28, 5/2, Mod(2,7)]); [f] = mfbasis(mf); ? [G,chi] = znchar(mf); [G.mod, chi] %7 = [7, [2]~] ? [G,chi] = znchar(f); chi %8 = [28, [0, 2]~]
The library syntax is GEN znchar(GEN D).
Let G be attached to (ℤ/qℤ)* (as per
G = znstar(q, 1)) and chi be a Dirichlet character on
(ℤ/qℤ)* (see Section se:dirichletchar or ??character).
Return the conductor of chi:
? G = znstar(126000, 1); ? zncharconductor(G,11) \\ primitive %2 = 126000 ? zncharconductor(G,1) \\ trivial character, not primitive! %3 = 1 ? zncharconductor(G,1009) \\ character mod 5^3 %4 = 125
The library syntax is GEN zncharconductor(GEN G, GEN chi).
Let N = ∏p pep and a Dirichlet character χ, we have a decomposition χ = ∏p χp into character modulo N where the conductor of χp divides pep; it equals pep for all p if and only if χ is primitive.
Given a znstar G describing a group (ℤ/Nℤ)*, a Dirichlet
character chi and an integer Q, return ∏p | (Q,N) χp.
For instance, if Q = p is a prime divisor of N, the function returns
χp (as a character modulo N), given as a Conrey
character (t_COL).
? G = znstar(40, 1); ? G.cyc %2 = [4, 2, 2] ? chi = [2, 1, 1]; ? chi2 = znchardecompose(G, chi, 2) %4 = [1, 1, 0]~ ? chi5 = znchardecompose(G, chi, 5) %5 = [0, 0, 2]~ ? znchardecompose(G, chi, 3) %6 = [0, 0, 0]~ ? c = charmul(G, chi2, chi5) %7 = [1, 1, 2]~ \\ t_COL: in terms of Conrey generators ! ? znconreychar(G,c) %8 = [2, 1, 1] \\ t_VEC: in terms of SNF generators
The library syntax is GEN znchardecompose(GEN G, GEN chi, GEN Q).
Given a Dirichlet character χ on G = (ℤ/Nℤ)* (see
znchar), return the complex Gauss sum
g(χ,a) = ∑n = 1N χ(n) e(a n/N)
? [G,chi] = znchar(-3); \\ quadratic Gauss sum: I*sqrt(3) ? znchargauss(G,chi) %2 = 1.7320508075688772935274463415058723670*I ? [G,chi] = znchar(5); ? znchargauss(G,chi) \\ sqrt(5) %2 = 2.2360679774997896964091736687312762354 ? G = znstar(300,1); chi = [1,1,12]~; ? znchargauss(G,chi) / sqrt(300) - exp(2*I*Pi*11/25) \\ = 0 %4 = 2.350988701644575016 E-38 + 1.4693679385278593850 E-39*I ? lfuntheta([G,chi], 1) \\ = 0 %5 = -5.79[...] E-39 - 2.71[...] E-40*I
The library syntax is GEN znchargauss(GEN G, GEN chi, GEN a = NULL, long bitprec).
Let G be attached to (ℤ/qℤ)* (as per G = znstar(q,1))
and let chi be a Dirichlet character on (ℤ/qℤ)*, given by
* a t_VEC: a standard character on bid.gen,
* a t_INT or a t_COL: a Conrey index in (ℤ/qℤ)* or its
Conrey logarithm;
see Section se:dirichletchar or ??character.
Let N be a multiple of q, return the character modulo N extending
chi. As usual for arithmetic functions, the new modulus N can be
given as a t_INT, via a factorization matrix or a pair
[N, factor(N)], or by znstar(N,1).
? G = znstar(4, 1); ? chi = znconreylog(G,1); \\ trivial character mod 4 ? zncharinduce(G, chi, 80) \\ now mod 80 %3 = [0, 0, 0]~ ? zncharinduce(G, 1, 80) \\ same using directly Conrey label %4 = [0, 0, 0]~ ? G2 = znstar(80, 1); ? zncharinduce(G, 1, G2) \\ same %4 = [0, 0, 0]~ ? chi = zncharinduce(G, 3, G2) \\ extend the nontrivial character mod 4 %5 = [1, 0, 0]~ ? [G0,chi0] = znchartoprimitive(G2, chi); ? G0.mod %7 = 4 ? chi0 %8 = [1]~
Here is a larger example:
? G = znstar(126000, 1); ? label = 1009; ? chi = znconreylog(G, label) %3 = [0, 0, 0, 14, 0]~ ? [G0,chi0] = znchartoprimitive(G, label); \\ works also with 'chi' ? G0.mod %5 = 125 ? chi0 \\ primitive character mod 5^3 attached to chi %6 = [14]~ ? G0 = znstar(N0, 1); ? zncharinduce(G0, chi0, G) \\ induce back %8 = [0, 0, 0, 14, 0]~ ? znconreyexp(G, %) %9 = 1009
The library syntax is GEN zncharinduce(GEN G, GEN chi, GEN N).
Let G be attached to (ℤ/Nℤ)* (as per G = znstar(N,1))
and let chi be a Dirichlet character on (ℤ/Nℤ)*, given by
* a t_VEC: a standard character on G.gen,
* a t_INT or a t_COL: a Conrey index in (ℤ/qℤ)* or its
Conrey logarithm;
see Section se:dirichletchar or ??character.
Return 1 if and only if chi(-1) = -1 and 0 otherwise.
? G = znstar(8, 1); ? zncharisodd(G, 1) \\ trivial character %2 = 0 ? zncharisodd(G, 3) %3 = 1 ? chareval(G, 3, -1) %4 = 1/2
The library syntax is long zncharisodd(GEN G, GEN chi).
Let G be attached to (ℤ/Nℤ)* (as per G = znstar(N,1))
and let chi be a Dirichlet character on (ℤ/Nℤ)*, given by
* a t_VEC: a standard character on bid.gen,
* a t_INT or a t_COL: a Conrey index in (ℤ/qℤ)* or its
Conrey logarithm;
see Section se:dirichletchar or ??character.
If flag = 0, return the discriminant D if chi is real equal to the
Kronecker symbol (D/.) and 0 otherwise. The discriminant D is
fundamental if and only if chi is primitive.
If flag = 1, return the fundamental discriminant attached to the corresponding primitive character.
? G = znstar(8,1); CHARS = [1,3,5,7]; \\ Conrey labels ? apply(t->znchartokronecker(G,t), CHARS) %2 = [4, -8, 8, -4] ? apply(t->znchartokronecker(G,t,1), CHARS) %3 = [1, -8, 8, -4]
The library syntax is GEN znchartokronecker(GEN G, GEN chi, long flag).
Let G be attached to (ℤ/qℤ)* (as per
G = znstar(q, 1)) and chi be a Dirichlet character on
(ℤ/qℤ)*, of conductor q0 | q.
? G = znstar(126000, 1); ? [G0,chi0] = znchartoprimitive(G,11) ? G0.mod %3 = 126000 ? chi0 %4 = 11 ? [G0,chi0] = znchartoprimitive(G,1);\\ trivial character, not primitive! ? G0.mod %6 = 1 ? chi0 %7 = []~ ? [G0,chi0] = znchartoprimitive(G,1009) ? G0.mod %4 = 125 ? chi0 %5 = [14]~
Note that znconreyconductor is more efficient since
it can return χ0 and its conductor q0 without needing to
initialize G0. The price to pay is a more cryptic format and the need to
initalize G0 later, but that needs to be done only once for all characters
with conductor q0.
The library syntax is GEN znchartoprimitive(GEN G, GEN chi).
Given a znstar G attached to (ℤ/qℤ)* (as per
G = znstar(q,1)), this function returns the Dirichlet character
attached to m ∈ (ℤ/qℤ)* via Conrey's logarithm, which
establishes a "canonical" bijection between (ℤ/qℤ)* and its dual.
Let q = ∏p pep be the factorization of q into distinct primes. For all odd p with ep > 0, let gp be the element in (ℤ/qℤ)* which is
* congruent to 1 mod q/pep,
* congruent mod pep to the smallest positive integer that generates (ℤ/p2ℤ)*.
For p = 2, we let g4 (if 2e2 ≥ 4) and g8 (if furthermore (2e2 ≥ 8) be the elements in (ℤ/qℤ)* which are
* congruent to 1 mod q/2e2,
* g4 = -1 mod 2e2,
* g8 = 5 mod 2e2.
Then the gp (and the extra g4 and g8 if 2e2 ≥ 2) are
independent generators of (ℤ/qℤ)*, i.e. every m in (ℤ/qℤ)*
can be written uniquely as ∏p gpmp, where mp is defined
modulo the
order op of gp and p ∈ Sq, the set of prime divisors of q
together with 4 if 4 | q and 8 if 8 | q. Note that the gp
are in general not SNF generators as produced by znstar whenever
ω(q) ≥ 2, although their number is the same. They however allow
to handle the finite abelian group (ℤ/qℤ)* in a fast and elegant way.
(Which unfortunately does not generalize to ray class groups or Hecke
characters.)
The Conrey logarithm of m is the vector (mp)p ∈ S_{q}, obtained
via znconreylog. The Conrey character χq(m,.) attached to
m mod q maps
each gp, p ∈ Sq to e(mp / op), where e(x) = exp(2iπ x).
This function returns the Conrey character expressed in the standard PARI
way in terms of the SNF generators G.gen.
? G = znstar(8,1);
? G.cyc
%2 = [2, 2] \\ Z/2 x Z/2
? G.gen
%3 = [7, 3]
? znconreychar(G,1) \\ 1 is always the trivial character
%4 = [0, 0]
? znconreychar(G,2) \\ 2 is not coprime to 8 !!!
*** at top-level: znconreychar(G,2)
*** ^ — — — — — --
*** znconreychar: elements not coprime in Zideallog:
2
8
*** Break loop: type 'break' to go back to GP prompt
break>
? znconreychar(G,3)
%5 = [0, 1]
? znconreychar(G,5)
%6 = [1, 1]
? znconreychar(G,7)
%7 = [1, 0]
We indeed get all 4 characters of (ℤ/8ℤ)*.
For convenience, we allow to input the Conrey logarithm of m instead of m:
? G = znstar(55, 1); ? znconreychar(G,7) %2 = [7, 0] ? znconreychar(G, znconreylog(G,7)) %3 = [7, 0]
The library syntax is GEN znconreychar(GEN G, GEN m).
Let G be attached to (ℤ/qℤ)* (as per
G = znstar(q, 1)) and chi be a Dirichlet character on
(ℤ/qℤ)*, given by
* a t_VEC: a standard character on bid.gen,
* a t_INT or a t_COL: a Conrey index in (ℤ/qℤ)* or its
Conrey logarithm;
see Section se:dirichletchar or ??character.
Return the conductor of chi, as the t_INT bid.mod
if chi is primitive, and as a pair [N, faN] (with faN the
factorization of N) otherwise.
If chi0 is present, set it to the Conrey logarithm of the attached
primitive character.
? G = znstar(126000, 1); ? znconreyconductor(G,11) \\ primitive %2 = 126000 ? znconreyconductor(G,1) \\ trivial character, not primitive! %3 = [1, matrix(0,2)] ? N0 = znconreyconductor(G,1009, &chi0) \\ character mod 5^3 %4 = [125, Mat([5, 3])] ? chi0 %5 = [14]~ ? G0 = znstar(N0, 1); \\ format [N,factor(N)] accepted ? znconreyexp(G0, chi0) %7 = 9 ? znconreyconductor(G0, chi0) \\ now primitive, as expected %8 = 125
The group G0 is not computed as part of
znconreyconductor because it needs to be computed only once per
conductor, not once per character.
The library syntax is GEN znconreyconductor(GEN G, GEN chi, GEN *chi0 = NULL).
Given a znstar G attached to (ℤ/qℤ)* (as per
G = znstar(q, 1)), this function returns the Conrey exponential of
the character chi: it returns the integer
m ∈ (ℤ/qℤ)* such that znconreylog(G, m) is chi.
The character chi is given either as a
* t_VEC: in terms of the generators G.gen;
* t_COL: a Conrey logarithm.
? G = znstar(126000, 1) ? znconreylog(G,1) %2 = [0, 0, 0, 0, 0]~ ? znconreyexp(G,%) %3 = 1 ? G.cyc \\ SNF generators %4 = [300, 12, 2, 2, 2] ? chi = [100, 1, 0, 1, 0]; \\ some random character on SNF generators ? znconreylog(G, chi) \\ in terms of Conrey generators %6 = [0, 3, 3, 0, 2]~ ? znconreyexp(G, %) \\ apply to a Conrey log %7 = 18251 ? znconreyexp(G, chi) \\ ... or a char on SNF generators %8 = 18251 ? znconreychar(G,%) %9 = [100, 1, 0, 1, 0]
The library syntax is GEN znconreyexp(GEN G, GEN chi).
Given a znstar attached to (ℤ/qℤ)* (as per
G = znstar(q,1)), this function returns the Conrey logarithm of
m ∈ (ℤ/qℤ)*.
Let q = ∏p pep be the factorization of q into distinct primes, where we assume e2 = 0 or e2 ≥ 2. (If e2 = 1, we can ignore 2 from the factorization, as if we replaced q by q/2, since (ℤ/qℤ)* ~ (ℤ/(q/2)ℤ)*.)
For all odd p with ep > 0, let gp be the element in (ℤ/qℤ)* which is
* congruent to 1 mod q/pep,
* congruent mod pep to the smallest positive integer that generates (ℤ/p2ℤ)*.
For p = 2, we let g4 (if 2e2 ≥ 4) and g8 (if furthermore (2e2 ≥ 8) be the elements in (ℤ/qℤ)* which are
* congruent to 1 mod q/2e2,
* g4 = -1 mod 2e2,
* g8 = 5 mod 2e2.
Then the gp (and the extra g4 and g8 if 2e2 ≥ 2) are
independent generators of ℤ/qℤ*, i.e. every m in (ℤ/qℤ)* can be
written uniquely as ∏p gpmp, where mp is defined modulo
the order op of gp and p ∈ Sq, the set of prime divisors of
q together with 4 if 4 | q and 8 if 8 | q. Note that the
gp
are in general not SNF generators as produced by znstar whenever
ω(q) ≥ 2, although their number is the same. They however allow
to handle the finite abelian group (ℤ/qℤ)* in a fast and elegant way.
(Which unfortunately does not generalize to ray class groups or Hecke
characters.)
The Conrey logarithm of m is the vector (mp)p ∈ S_{q}. The inverse
function znconreyexp recovers the Conrey label m from a character.
? G = znstar(126000, 1);
? znconreylog(G,1)
%2 = [0, 0, 0, 0, 0]~
? znconreyexp(G, %)
%3 = 1
? znconreylog(G,2) \\ 2 is not coprime to modulus !!!
*** at top-level: znconreylog(G,2)
*** ^ — — — — — --
*** znconreylog: elements not coprime in Zideallog:
2
126000
*** Break loop: type 'break' to go back to GP prompt
break>
? znconreylog(G,11) \\ wrt. Conrey generators
%4 = [0, 3, 1, 76, 4]~
? log11 = ideallog(,11,G) \\ wrt. SNF generators
%5 = [178, 3, -75, 1, 0]~
For convenience, we allow to input the ordinary discrete log of m,
ideallog(,m,bid), which allows to convert discrete logs
from bid.gen generators to Conrey generators.
? znconreylog(G, log11) %7 = [0, 3, 1, 76, 4]~
We also allow a character (t_VEC) on bid.gen and
return its representation on the Conrey generators.
? G.cyc %8 = [300, 12, 2, 2, 2] ? chi = [10,1,0,1,1]; ? znconreylog(G, chi) %10 = [1, 3, 3, 10, 2]~ ? n = znconreyexp(G, chi) %11 = 84149 ? znconreychar(G, n) %12 = [10, 1, 0, 1, 1]
The library syntax is GEN znconreylog(GEN G, GEN m).
Coppersmith's algorithm. N being an integer and P ∈ ℤ[t], finds in polynomial time in log(N) and d = deg(P) all integers x with |x| ≤ X such that gcd(N, P(x)) ≥ B. This is a famous application of the LLL algorithm meant to help in the factorization of N. Notice that P may be reduced modulo Nℤ[t] without affecting the situation. The parameter X must not be too large: assume for now that the leading coefficient of P is coprime to N, then we must have d log X log N < log2 B, i.e., X < N1/d when B = N. Let now P0 be the gcd of the leading coefficient of P and N. In applications to factorization, we should have P0 = 1; otherwise, either P0 = N and we can reduce the degree of P, or P0 is a non trivial factor of N. For completeness, we nevertheless document the exact conditions that X must satisfy in this case: let p := logN P0, b := logN B, x := logN X, then
* either p ≥ d / (2d-1) is large and we must have x d < 2b - 1;
* or p < d / (2d-1) and we must have both p < b < 1 - p + p/d and x(d + p(1-2d)) < (b - p)2. Note that this reduces to x d < b2 when p = 0, i.e., the condition described above.
Some x larger than X may be returned if you are
very lucky. The routine runs in polynomial time in log N and d
but the smaller B, or the larger X, the slower.
The strength of Coppersmith method is the ability to find roots modulo a
general composite N: if N is a prime or a prime power,
polrootsmod or polrootspadic will be much faster.
We shall now present two simple applications. The first one is finding nontrivial factors of N, given some partial information on the factors; in that case B must obviously be smaller than the largest nontrivial divisor of N.
setrand(1); \\ to make the example reproducible [a,b] = [10^30, 10^31]; D = 20; p = randomprime([a,b]); q = randomprime([a,b]); N = p*q; \\ assume we know 0) p | N; 1) p in [a,b]; 2) the last D digits of p p0 = p % 10^D; ? L = zncoppersmith(10^D*x + p0, N, b \ 10^D, a) time = 1ms. %6 = [738281386540] ? gcd(L[1] * 10^D + p0, N) == p %7 = 1
and we recovered p, faster than by trying all possibilities x < 1011.
The second application is an attack on RSA with low exponent, when the message x is short and the padding P is known to the attacker. We use the same RSA modulus N as in the first example:
setrand(1); P = random(N); \\ known padding e = 3; \\ small public encryption exponent X = floor(N^0.3); \\ N^(1/e - epsilon) x0 = random(X); \\ unknown short message C = lift( (Mod(x0,N) + P)^e ); \\ known ciphertext, with padding P zncoppersmith((P + x)^3 - C, N, X) \\ result in 244ms. %14 = [2679982004001230401] ? %[1] == x0 %15 = 1
We guessed an integer of the order of 1018, almost instantly.
The library syntax is GEN zncoppersmith(GEN P, GEN N, GEN X, GEN B = NULL).
This functions allows two distinct modes of operation depending on g:
* if g is the output of znstar (with initialization),
we compute the discrete logarithm of x with respect to the generators
contained in the structure. See ideallog for details.
* else g is an explicit element in (ℤ/Nℤ)*, we compute the discrete logarithm of x in (ℤ/Nℤ)* in base g. The rest of this entry describes the latter possibility.
The result is [] when x is not a power of g, though the function may also enter an infinite loop in this case.
If present, o represents the multiplicative order of g, see
Section se:DLfun; the preferred format for this parameter is
[ord, factor(ord)], where ord is the order of g.
This provides a definite speedup when the discrete log problem is simple:
? p = nextprime(10^4); g = znprimroot(p); o = [p-1, factor(p-1)]; ? for(i=1,10^4, znlog(i, g, o)) time = 163 ms. ? for(i=1,10^4, znlog(i, g)) time = 200 ms. \\ a little slower
The result is undefined if g is not invertible mod N or if the supplied order is incorrect.
This function uses
* a combination of generic discrete log algorithms (see below).
* in (ℤ/Nℤ)* when N is prime: a linear sieve index calculus method, suitable for N < 1050, say, is used for large prime divisors of the order.
The generic discrete log algorithms are:
* Pohlig-Hellman algorithm, to reduce to groups of prime order q, where q | p-1 and p is an odd prime divisor of N,
* Shanks baby-step/giant-step (q < 232 is small),
* Pollard rho method (q > 232).
The latter two algorithms require O(sqrt{q}) operations in the group on average, hence will not be able to treat cases where q > 1030, say. In addition, Pollard rho is not able to handle the case where there are no solutions: it will enter an infinite loop.
? g = znprimroot(101) %1 = Mod(2,101) ? znlog(5, g) %2 = 24 ? g^24 %3 = Mod(5, 101) ? G = znprimroot(2 * 101^10) %4 = Mod(110462212541120451003, 220924425082240902002) ? znlog(5, G) %5 = 76210072736547066624 ? G^% == 5 %6 = 1 ? N = 2^4*3^2*5^3*7^4*11; g = Mod(13, N); znlog(g^110, g) %7 = 110 ? znlog(6, Mod(2,3)) \\ no solution %8 = []
For convenience, g is also allowed to be a p-adic number:
? g = 3+O(5^10); znlog(2, g) %1 = 1015243 ? g^% %2 = 2 + O(5^10)
The library syntax is GEN znlog0(GEN x, GEN g, GEN o = NULL).
The function
GEN znlog(GEN x, GEN g, GEN o) is also available
x must be an integer mod n, and the
result is the order of x in the multiplicative group (ℤ/nℤ)*. Returns
an error if x is not invertible.
The parameter o, if present, represents a nonzero
multiple of the order of x, see Section se:DLfun; the preferred format for
this parameter is [ord, factor(ord)], where ord = eulerphi(n)
is the cardinality of the group.
The library syntax is GEN znorder(GEN x, GEN o = NULL).
Returns a primitive root (generator) of (ℤ/nℤ)*, whenever this latter group is cyclic (n = 4 or n = 2pk or n = pk, where p is an odd prime and k ≥ 0). If the group is not cyclic, the function will raise an exception. If n is a prime power, then the smallest positive primitive root is returned. This may not be true for n = 2pk, p odd.
Note that this function requires factoring p-1 for p as above, in order to determine the exact order of elements in (ℤ/nℤ)*: this is likely to be costly if p is large.
The library syntax is GEN znprimroot(GEN n).
Gives the structure of the multiplicative group (ℤ/nℤ)*. The output G depends on the value of flag:
* flag = 0 (default), an abelian group structure [h,d,g],
where h = φ(n) is the order (G.no), d (G.cyc)
is a k-component row-vector d of integers di such that di > 1,
di | di-1 for i ≥ 2 and
(ℤ/nℤ)* ~ ∏i = 1k (ℤ/diℤ),
and g (G.gen) is a k-component row vector giving generators of
the image of the cyclic groups ℤ/diℤ.
* flag = 1 the result is a bid structure;
this allows computing discrete logarithms using znlog (also in the
noncyclic case!).
? G = znstar(40) %1 = [16, [4, 2, 2], [Mod(17, 40), Mod(21, 40), Mod(11, 40)]] ? G.no \\ eulerphi(40) %2 = 16 ? G.cyc \\ cycle structure %3 = [4, 2, 2] ? G.gen \\ generators for the cyclic components %4 = [Mod(17, 40), Mod(21, 40), Mod(11, 40)] ? apply(znorder, G.gen) %5 = [4, 2, 2]
For user convenience, we define znstar(0) as
[2, [2], [-1]], corresponding to ℤ*, but flag = 1 is not
implemented in this trivial case.
The library syntax is GEN znstar0(GEN n, long flag).
Finds a minimal set of generators for the subgroup of (ℤ/fℤ)*
given by a vector (or vectorsmall) H of length f:
for 1 ≤ a ≤ f, H[a] is 1 or 0 according as a ∈ HF
or a ∉ HF. In most PARI functions, subgroups of an abelian group
are given as HNF left-divisors of a diagonal matrix, representing the
discrete logarithms of the subgroup generators in terms of a fixed
generators for the group cyclic components. The present function
allows to convert an enumeration of the subgroup elements to this
representation as follows:
? G = znstar(f, 1); ? v = znsubgroupgenerators(H); ? subHNF(G, v) = mathnfmodid(Mat([znlog(h, G) | h<-v]), G.cyc);
The function subHNF can be applied to any
elements of (ℤ/fℤ)*, yielding the subgroup they generate, but using
znsubgroupgenerators first allows to reduce the number of discrete
logarithms to be computed.
For example, if H = {1,4,11,14} ⊂ (ℤ/15ℤ)×, then we have
? f = 15; H = vector(f); H[1]=H[4]=H[11]=H[14] = 1; ? v = znsubgroupgenerators(H) %2 = [4, 11] ? G = znstar(f, 1); G.cyc %3 = [4, 2] ? subHNF(G, v) %4 = [2 0] [0 1] ? subHNF(G, [1,4,11,14]) %5 = [2 0] [0 1]
This function is mostly useful when f is large
and H has small index: if H has few elements, one may just use
subHNF directly on the elements of H. For instance, let
K = ℚ(ζp, sqrt{m}) ⊂ L = ℚ(ζf), where p is
a prime, sqrt{m} is a quadratic number and f is the conductor of the
abelian extension K/ℚ. The following GP script creates H as the Galois
group of L/K, as a subgroub of (ℤ/fZ)*:
HK(m, p, flag = 0)=
{ my(d = quaddisc(m), f = lcm(d, p), H);
H = vectorsmall(f, a, a % p == 1 && kronecker(d,a) > 0);
[f, znsubgroupgenerators(H,flag)];
}
? [f, v] = HK(36322, 5)
time = 193 ms.
%1 = [726440, [41, 61, 111, 131]]
? G = znstar(f,1); G.cyc
%2 = [1260, 12, 2, 2, 2, 2]
? A = subHNF(G, v)
%3 =
[2 0 1 1 0 1]
[0 4 0 0 0 2]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
\\ Double check
? p = 5; d = quaddisc(36322);
? w = select(a->a % p == 1 && kronecker(d,a) > 0, [1..f]); #w
time = 133 ms.
%5 = 30240 \\ w enumerates the elements of H
? subHNF(G, w) == A \\ same result, about twice slower
time = 242 ms.
%6 = 1
This shows that K = ℚ(sqrt{36322},ζ5) is contained in
ℚ(ζ726440) and H = <41, 61, 111, 131 >.
Note that H = <41><61><111 >
<131> is not a direct product. If flag = 1, then the function
finds generators which decompose H to direct factors:
? HK(36322, 5, 1) %3 = [726440, [41, 31261, 324611, 506221]]
This time
H = <41>x <31261>x
<324611 >x <506221 >.
The library syntax is GEN znsubgroupgenerators(GEN H, long flag).