Primes and polygonal numbers

TL;DR

This paper investigates the distribution of primes generated by linear combinations of polygonal numbers, showing infinite primes under certain conditions and the existence of long prime progressions of quadratic forms.

Contribution

It establishes conditions for infinitely many primes from linear combinations of polygonal numbers and demonstrates long arithmetic progressions among primes of quadratic forms.

Findings

Infinite primes from linear combinations of polygonal numbers with coprime coefficients.

Convergence of reciprocal sums of such primes unless specific quadratic forms.

Existence of arbitrarily long arithmetic progressions among primes of the form am^2+bn^2.

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 $am^2+bn^2$.