[seqfan] Re: Integer points on hyperelliptic curve

[Noam Elkies]

Artur <grafix at csl.pl> asks:

> Who know how to find integer points on hyperelliptic curve
> y^2=x^5+2869
> or prove that these points doesn't exists.
> I'm asking about programme and procedure
> (MAGMA, CASH, MAPLE, MWRANK others)

Michael Stoll's "ratpoints" program takes only a few seconds to verify
that there are no integer points up to x=10^10 using an intelligent
(sieving) exhaustive search.

It also indicates that there are no rational solutions with x=m/n and
max(|m|,|n|) < 10^6, which strongly suggests that there are no such points
(outside of the "point at infinity").

Nowadays when a curve of genus 2 has no rational points, or only a few,
this can often be proved by combining descent and Chabauty techniques.
I can't do this; ask Nils Bruin.

Sincerely,
--Noam D. Elkies