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.*



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> **Question.** Is there any reasoning or heuristic to get an idea about the veracity of previous conjecture? **Many thanks.**
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**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:
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[1] The article [*Friedlander–Iwaniec theorem*](https://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem) from the encyclopedia Wikipedia.



Are there infinitely many primes of the form $\frac{3a^2-a}{2}+b^4$?
What can be done, which is already remarked in the cited paper by Friedlander and Iwaniec, is that for any positive definite binary quadratic form $f$ which has no local obstructions, there ought to exist infinitely many pairs $(a,b) \in \mathbb{Z}^2$ such that $f(a,b^2)$ is prime. Even though this is widely believed, the technical weight of the F-I paper has thus far prevented anyone from writing this up formally and publishing it. 


As remarked by Wojowu, the crucial property of the polynomial $x^2 + y^4$ is that it factors over a number field. This additional structure enables one to obtain satisfactory bilinear sum estimates, which Friedlander and Iwaniec shows in a companion paper to the above cited paper is all that's needed for Bombieri's sieve to produce primes. So far, the existence of a norm form structure (the only polynomials we know that 'obviously' produce infinitely many primes) is necessary to obtain such bilinear sum estimates.


Aside from the Friedlander-Iwaniec paper mentioned, there are three seminal papers that run along these lines:


1) D.R. Heath-Brown, *Primes represented by $x^3 + 2y^3$*, Acta Math., (1) **186** (2001), 1-84.


2) J. Maynard, *Primes represented by incomplete norm forms*, <https://arxiv.org/abs/1507.05080>.


3) D.R. Heath-Brown, X. Li, *Prime values of $a^2 + p^4$*, Invent. Math (2) **208**, 441-499.


In general, one would expect that any polynomial that is not reducible over $\mathbb{Q}$ and for which there are no obvious obstructions would represent infinitely many primes. This is considered well beyond reach of current methods. 


