Show that for any prime p, there are integers $x, y$ such that $p| x^2+y^2+1$

Question

Show that for any prime p, there are integers $x, y$ such that $p\mid x^2+y^2+1$

Answer 1

Hint: Consider an odd prime $p$.

Show that the residues of $x^2$ modulo $p$ for $0\le x\le (p-1)/2$ are distinct.

Similarly,the residues of $-y^2-1$ modulo $p$ for $0\le y\le (p-1)/2$ are distinct. Then apply pigeonhole principle.