Primes of the form $ax+by$

Abstract

For two coprime positive integers $a,b$, let $T(a,b)=\{ ax+by : x,y\in \mathbb{Z}_{\ge 0} \} $ and let $s(a,b)=ab-a-b$. It is well known that all integers which are greater than $s(a,b)$ are in $T(a,b)$. Let $\pi (a, b)$ be the number of primes in $T(a,b)$ which are less than or equal to $s(a,b)$. It is easy to see that $\pi (2, 3)=0$ and $\pi (2, b)=1$ for all odd integers $b\ge 5$. In this paper, we prove that if $b>a\ge 3$ with $\gcd (a, b)=1$, then $\pi (a, b)>0.005 s(a,b)/\log s(a,b)$. We conjecture that $\frac{13}{66}\pi (s(a,b))\le \pi (a, b)\le \frac 12\pi (s(a,b))$ for all $b>a\ge 3$ with $\gcd (a, b)=1$.