# “Last Fermat's Theorem” modulo m

The last Fermat’s Theorem is a claim about the non-existence of non-trivial integer solution for $X^n+Y^n=Z^n$ for $n\in \mathbb N$, $n\ge 3$.

However, given $m\in \mathbb N$, we can investigate the integer solutions for $X^n+Y^n\equiv Z^n \mod m$ for all $n\in \mathbb N$ with the restriciton that $X,Y,Z\not\equiv0\mod m$.

It seems to me that we always have solutions in this case, but I did no find any reference or exposition about this.

Is it “relevant” to think on it? Have this question appeared elsewhere?

#### Solutions Collecting From Web of "“Last Fermat's Theorem” modulo m"

You are correct. For sufficiently large primes $p$ and any $n \geq 1$, there are always nontrivial solutions to
$$X^n + Y^n \equiv Z^n \pmod p.$$
Schur first proved this in 1916, in his paper — Schur, I. “Über die Kongruenz x^m+y^m=z^m (mod p).” Jahresber. Deutsche Math.-Verein. 25, 114-116, 1916.

You might think it meta-wise clear that there are solutions mod $p$ for every $p$, as otherwise the problem wouldn’t really be so hard. [Or perhaps you might think it’s very hard to find a $p$ for which there are no solutions, and that was the bottleneck.]

As mixedmath says, this is true for sufficiently large prime $m$. This result is due to Schur. See this blog post for an exposition.