Primes and polygonal numbers

Abstract

A linear combination $aT_r(m)+bT_s(n)$ of an \mbox{$r$-gonal} number $T_r(m)$ and an $s$-gonal number $T_s(n)$ with mutually coprime positive integer coefficients $a$ and $b$ produces infinitely many primes as $m$ and~$n$ varies over the natural numbers, whereas the sum of the reciprocals of such primes converges unless $T_r(m)=m^2$ and $T_s(n)=n^2$. For each pair of coprime positive integers $a$ and $b$, there are arbitrary long arithmetic progressions among the primes of the form $am^2+bn^2$.