I was inspired from a theorem due to Iwaniec and Friedlander, see [1], to ask the following conjecuture involving integers.
Conjecture. There are infinitely many prime numbers of the form $$\frac{3a^2-a}{2}+b^4\tag{1}$$ where $a$ and $b$ run over positive integers.
Question. Is there any reasoning or heuristic to get an idea about the veracity of previous conjecture? Many thanks.
Remarks and motivation. The first few terms of previous expression $(1)$ are $$2, 13, 17, 23, 67, 71,\ldots$$ I believe that this sequence isn't in the OEIS. My motivation to ask about previous conjecture was to propose a variation of the work by Friedlander and Iwaniec, involving the pentagonal numbers (a difererent set of polygonal numbers) instead of perfect squares $a^2$.
References:
[1] The article Friedlander–Iwaniec theorem from the encyclopedia Wikipedia.