cino hilliard on Thu, 18 Jun 2009 10:59:47 +0200

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 Elliptic curve x^3 - y^2 = p

 Hi , I want to find the number of solutions of  the elliptic curve, x^3 - y^2 = p for various p = 7, 431, 503, etc   I have been using brute force in a Pari script below testing for solutions.  diffcubesq2(n,p) =               {               local(a,c=0,c2=0,j,k,y);               for(j=1,n,                for(k=1,n,                 y=j^3-k^2;                  if(y==p,                   c++;                   print(j","k","y);                    );                   );                  );                   c;                    }  diffcubesq2(10000,431) outputs 8,9,43111,30,43120,87,43130,163,43136,215,431138,1621,431150,1837,431 (03:14:10) gp > ##  ***   last result computed in 6mn, 57,969 ms.   My Pari code misses the last two solutions. It would have taken way too much time to get to y = 243836 anyway.   >From a example and link posted by David Broadhurst in another forum,   http://magma.maths.usyd.edu.au/calc/   I tried using the Magma applet to compute the elliptic curve.This gets all solutions in a fraction of the time.   Does anyone have Pari script with elliptic curve functions that will do something like this?     Examples: p = 7 has 2 solutions, p = 431 has 9 Input: E := EllipticCurve([0, -7]);Q, reps := IntegralPoints(E);Q; Output: Magma V2.15-3     Thu Jun 18 2009 17:24:22    [Seed = 2348296821]   ------------------------------------- [ (2 : 1 : 1), (32 : -181 : 1) ] Total time: 0.350 seconds, Total memory usage: 136.95MB Input: E := EllipticCurve([0, -431]);Q, reps := IntegralPoints(E);Q;   Output: Magma V2.15-3     Thu Jun 18 2009 17:33:07    [Seed = 3601166652]   ------------------------------------- [ (8 : 9 : 1), (20 : -87 : 1), (11 : -30 : 1), (30 : -163 : 1), (36 : 215 : 1), (138 : 1621 : 1), (150 : 1837 : 1), (575 : 13788 : 1), (3903 : -243836 : 1) ] Total time: 1.050 seconds, Total memory usage: 137.04MBThanks, Cino