Heath-Brown [Acta Math. 186(2001), 1-84] proved in 2001 that there are infinitely many primes of the form $x^3+2y^3$ with $x,y\in\mathbb Z^+=\{1,2,3,\ldots\}$.
In contrast, here I ask the following question.
Question 1. Are there infinitely many primes of the form $x^2+xy^2+y^3=x^2+(x+y)y^2$ with $x,y\in\mathbb Z^+$? Is it possible to provide a positive answer with a proof?
This question is motivated by my following conjecture (cf. http://oeis.org/A232174).
Conjecture 1. Any integer $n>1$ can be written as $x+y$ with $x,y\in\mathbb Z^+$ such that $x+ny$ and $x^2+ny^2$ are both prime.
This conjecture implies that Question 1 has a positive answer.
By a theorem of Heath-Brown and Moroz [Proc. London Math. Soc. 84(2002), 257-288], there are infinitely many primes of the form $x^3+(x+y)y^2$ with $x,y\in\mathbb Z^+$. Here I ask the following further question.
Question 2. Are there infinitely many primes of the form $p^3+(p+q)q^2$ with $p$ prime and $q\in\mathbb Z^+$?
This was motivated by my following conjecture (cf. http://oeis.org/A232186).
Conjecture 2. Any integer $n>2$ can be written as $p+q$ with $q\in\mathbb Z^+$ such that $p$ and $p^3+nq^2$ are both prime.
Conjecture 2 implies that Question 2 has a positive answer.
