Question 138
Let $n$ be a positive integer and $0 \le m \le n^2 - 1$. Prove that the total number of $m$ such that
$$
x^n + y^n \equiv m \pmod{n^2}
$$
has no integer solution, is at least $\frac{n(n-1)}{2}$.
$$
x^n + y^n \equiv m \pmod{n^2}
$$
has no integer solution, is at least $\frac{n(n-1)}{2}$.