According to this SE question (and the internet more generally), $2$ is a quartic residue modulo a prime $p\equiv 1\pmod{4}$ if and only if $p$ may be written in the form $a^2 + 64b^2$ for integers $a$ and $b$. Naively, this condition seems a bit unwieldy, and I would like some simple way of telling whether a prime satisfies it. Does such a condition exist?
Comparing the the quadratic case, $2$ is a quadratic residue if and only if $p \equiv \pm 1\pmod{8}$. This feels much nicer and more explicit to me, although I suspect the quartic case may be inherently nastier.