Problem

Source: Indonesian IMO TST 2011

Tags: modular arithmetic, induction, number theory unsolved, number theory



A prime number $p$ is a moderate number if for every $2$ positive integers $k > 1$ and $m$, there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \] If $q$ is the smallest moderate number, then determine the smallest prime $r$ which is not moderate and $q < r$.