are there infinitely many primes $p,q$ such that $pq=a^2+b^4$

Question

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 $4$.

https://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem is about primes rather than semiprimes — Will Jagy, Mar 07 '21 at 03:36

score 1 · Accepted Answer · answered Mar 07 '21 at 05:55

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.

are there infinitely many primes $p,q$ such that $pq=a^2+b^4$

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