       Re: integer solutions

• To: mathgroup at smc.vnet.net
• Subject: [mg30644] Re: [mg30635] integer solutions
• From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
• Date: Sat, 1 Sep 2001 01:08:16 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```As long as the modulus is not very large the simplest thing to do is to
use Solve. For example, in your case

DeleteCases[
({x -> #1, Solve[{y^2 == #1^3 - 2*#1 + 1, Modulus == 12}, y, Mode ->
Modular]} & ) /@ Range /. {Modulus -> 12, z_} -> z, {z_, {}},
Infinity]

Out=
{{x -> 1, {y -> 0, y -> 6}}, {x -> 4, {y -> 3, y -> 9}},

{x -> 6, {y -> 1, y -> 5, y -> 7, y -> 11}},

{x -> 9, {y -> 2, y -> 4, y -> 8, y -> 10}},

{x -> 10, {y -> 3, y -> 9}},

{x -> 12, {y -> 1, y -> 5, y -> 7, y -> 11}}}

Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/

On Saturday, September 1, 2001, at 08:58  AM, Adrian Spalka wrote:

> hi,
>
> how can i tell mathematica to compute all solutions of an equation in
> two
> variables in Fp?
>
> eg, to compute all solutions of
>
>     y^2=x^3-2*x+1     in F13, ie with x,y in [0,...,12]
>
> thanks.
>
>
>
>
>
>
>

```

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