If we suppose that we are interested in non-negative integers less than a prime $p$, is it possible to find more than two of these integers, such that, when squared, are all congruent modulo $p$?
In other words, can we find non-negative integers $x$, $y$, $z$, such that $0 \le x < y < z < p$ and $x^2 \equiv y^2 \equiv z^2 \bmod p$
I believe that we can't, but I'm not sure how to prove this.