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MathGroup Archive 2007

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Re: Integer Points on elliptic curve

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82430] Re: [mg82411] Integer Points on elliptic curve
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Sat, 20 Oct 2007 05:48:41 -0400 (EDT)
  • Reply-to: hanlonr at cox.net

Brute force method:

eqn = a^2 + 1689 b^2 == c^3;

soln = Select[
  Flatten[Table[{a, b, (eqn[[1]])^(1/3)}, {a, 0, 2000}, {b, 0, 80}], 
   1], IntegerQ[Last[#]] &]

{{0, 0, 0}, {1, 0, 1}, {8, 0, 4}, 
   {27, 0, 9}, {64, 0, 16}, 
   {125, 0, 25}, {144, 3, 33}, 
   {205, 32, 121}, {216, 0, 36}, 
   {241, 10, 61}, {343, 0, 49}, 
   {434, 2, 58}, {512, 0, 64}, 
   {729, 0, 81}, {1000, 0, 100}, 
   {1152, 24, 132}, {1276, 29, 145}, 
   {1331, 0, 121}, {1366, 14, 130}, 
   {1654, 59, 205}, {1728, 0, 144}, 
   {1928, 80, 244}}


Bob Hanlon

---- Artur <grafix at csl.pl> wrote: 
> Who know how find by Mathematica integer solutions of elliptic equation:
> a^2+1689b^2=c^3
> one can be find by procedure:
> Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 1689])) ]);
> but these procedure work only for b=0
> 
> BEST WISHES
> ARTUR JASINSKI
> 
> 
> 
> 



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