are there infinitely many primes $p,q$ such that $pq=a^2+b^4$ Are there infinitely many primes $p<q$, $p,q\neq 2,3$ such that $pq=a^2+b^4$ where $a,b\in \mathbb{Z}$ ? I've no idea if this is a very easy or very hard question. Any known result about this ?
Thank you for your comments !
EDIT : using mod 4 arguments you can derive very easily conditions on a and b. Also using Gaussian integers, this product pq boils down to the product of 4 gaussian irreducible elements. Note that p,q must be congruent to 1 mod 4.

The answer should certainly be yes.
For example, with $b=1$ and $a = 2 + 5 x$, we'll have $a^2 + b^4 = 5 (1+4x+5x^2)$, and Bunyakovsky's conjecture implies there are infinitely many integers $x$ for which $1+4x+5x^2$ is prime.
However, no proof of this is known.
