Is there any way to find non-trivial solutions to the equation $x^2 + y^2 - x \equiv 0 \mod{N}$? There are clearly several trivial solutions, for example $(x, y) = (0, 0), (1, 0), (2^{-1}, 2^{-1}), (2^{-1}, -2^{-1})$. By trivial solution, I mean one that holds for all choices of $N$ and hence do not tell us anything about the structure of any specific $N$. In this problem, assume the factorization of $N$ is not known.