Fermat's last theorem fails in $\mathbb{Z}/p\mathbb Z$ for $p$ sufficiently large

Question

Statement

For any $n, \;x^n+y^n=z^n$ has non-trivial solutions in $\mathbb{Z}/p\mathbb Z$ for all but finitely many $p$.

I remember seeing this problem on an first year undergraduate problem sheet, but never succeeded in solving it. I cannot find an elementary solution on the internet though: does anyone know of one?

score 5 · Answer 1 · answered Mar 03 '14 at 14:48

5

Not sure what you consider to be elementary but it can be solved with Schur's theorem. See:

http://math.mit.edu/~fox/MAT307-lecture05.pdf

It is theorem 4 in that paper.

answered Mar 03 '14 at 14:48

ploosu2

12,868

Looks elementary enough to me! I think this question should be closed so as to not clutter up the list of unanswered questions. The asker is gone... – Stefanie Jan 25 '19 at 07:17

Fermat's last theorem fails in $\mathbb{Z}/p\mathbb Z$ for $p$ sufficiently large

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