“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.