Are there infinitely many prime numbers $p$ such that $$p = x^4+y^4$$ for some $x,y \in \Bbb Z$ ? What if we only require $x,y \in \Bbb Q$ ?
I know that $p = a^2+b^2$ with $a,b \in \Bbb Q$ iff $p = a^2+b^2$ with $a,b \in \Bbb Z$ (Davenport-Cassels) iff $p=2$ or $p \equiv 1 \pmod 4$. But $a$ and $b$ might not be squares. What are some references about this problem, which explain what is (un)known?