If $(a,b)=1$ and $2|b$ then $p$ is prime and $p|a^2+b^2\implies p\not\equiv3\bmod4$.
Is there a similar statement at any other $k\in\Bbb N_{>2}$ that gives for $p\not\equiv(2^k-1)\bmod2^k$ but allows for $p\equiv i\bmod2^k$ where $i\in\{1,3,\dots,2^k-3\}$?
Is there a classification for infinitely many $k\in\Bbb N$?