For every odd prime $q \geq 3$, does there exist a prime $p < q$ and integers $x,y$ such that
$$\displaystyle x^2 + py^2 = q?$$
One can easily show that all primes $q \not \equiv -1 \pmod{3}$ can be written as such a sum. Indeed, if $q \equiv 1 \pmod{3}$ then $q$ is the norm of an element in the ring of Eisenstein integers, so there exist integers $x,y$ such that $q = x^2 + xy + y^2$. If $xy \equiv 0 \pmod{2}$, say $2 | y$, then we can write
$$\displaystyle q = \left(x + \frac{y}{2} \right)^2 + 3 \left(\frac{y}{2} \right)^2,$$
so we are done. Otherwise we must have $x,y$ are both odd. Now note that $q = x^2 + xy + y^2$ implies that
$$\displaystyle q = (-x - y)^2 + y(-x - y) + y^2,$$
and $x + y$ is even.
Playing similar games for other small primes of class number one gives that at least half of odd primes can be expressed in the form above.
