BxMO 2019 Problem 4

An integer $m>1$ is rich if for any positive integer $n$, there exist positive integers $x, y, z$ such that $n=m x^{2}-y^{2}-z^{2}$. An integer $m>1$ is poor if it is not rich. a) Find a poor integer. b) Find a rich integer. a) Solution I. We will show that $m=4$ is poor. If $y$ and $z$ are both even, we have $4 x^{2}-y^{2}-z^{2} \equiv$ $0-0-0=0(\bmod 4)$. If $y$ is even and $z$ is odd or the other way around, then $4 x^{2}-y^{2}-z^{2} \equiv$ $0-0-1 \equiv 3(\bmod 4)$. If $y$ and $z$ are both odd, we have $4 x^{2}-y^{2}-z^{2} \equiv 0-1-1 \equiv 2(\bmod 4)$. Hence it is impossible to write any integer $n \equiv 1(\bmod 4)$ as $n=m x^{2}-y^{2}-z^{2}$. So $m=4$ is poor.