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?