From: cet1-nospam@cam.ac.uk.invalid (Chris Thompson)
Subject: Re: Fermat equation
Date: 27 Nov 2000 17:12:43 GMT
Newsgroups: sci.math
Summary: [missing]

In article <975156936.20025.0.nnrp-09.c2ded8d9@news.demon.co.uk>,
Charles Matthews <charles@sabaki.demon.co.uk> wrote:
>
>Carl Johan Ragnarsson wrote
>
>>    x^n+y^n=z^n (mod p) looks to have solutions for most n and p by
>>inspection
>>    (with x,y,z=/=0). Is it true that it has solutions for all n and p
>>(well,
>>    all sufficiently large n and p)
>
>
>Yes, for n fixed it has solutions for all sufficiently large p.  A proof
>that covers this case can be found in Borevich-Shafarevich Number Theory,
>Ch.1.

In section 6.1 of Washington's "Introduction to Cyclotomic Fields" (GTM 83)
you'll find this used as an exercise in the use of Gauss and Jacobi sums.
The number of solutions in projective space over the finite field of q 
elements is bounded by

   | (#solutions) - (q+1) | < (n-1)(n-2) sqrt(q)

(and so is certainly non-zero for q large enough).

Those familiar with the application of algebraic geometry to number theory
will recognise this as a special case of a general theorem, the genus of the
curve x^n + y^n = z^n being (n-1)(n-2)/2.

Chris Thompson
Email: cet1 [at] cam.ac.uk