Suppose that we are given two positive integers $x$ and $y$ such that
$$x \mod p \leqslant y \mod p$$
for each prime number $p$. (Here, $x \mod p,\; y \mod p$ stand for the least non-negative residua.) Does it follow that $x = y$?
The problem is seemingly easy as we have to test $x,y$ against finitely many primes only. However, after several attempts I begin to wonder whether this is an open problem...
Note that it seems to be an open problem whether there is a prime number between a pair of squares (a reference would be appreciated), so the case where $x$ and $y$ are squares themselves is hard enough. However, it may well happen that this doesn't require such an argument.