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)

```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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