I am interested in primes and some of their varying characteristics which make them “qualitatively” different such as whether each prime generated by a certain type of Dirichlet arithmetic progressions belongs or not to the class of primes that are obtained from powerful order terms of that progression. Particularly, I wonder whether one can go beyond Dirichlet Theorem and establish that there is an infinite supply of this sort of primes in these progressions.
The case I am particularly interested is the case when d=1 and a=2k with k positive integer, but I would like to formulate the following Conjecture for the general case
For any two coprime positive integers a and d, there are infinite primes of the form $an^2m^3+d$ with n and m integers >=0
If true this would strengthen Dirichlet theorem on arithmetic progressions selecting only the terms of the progressions whose order is a powerful number (a number that if prime p divides it, $p^2$ also divides it).
I wonder about the likeliness of the conjecture (which intuitively seems high) and whether there is any proof I am still not aware of. I welcome estimates on the toughness of the proof, as well as any hints towards its development.