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 $\pi (a,b)>0.005s(a,b)/\log s(a,b)$. We conjecture that $\frac{13}{66}\pi (s(a,b))\le \pi (a,b)\le \frac{1}{2}\pi (s(a,b))$ for all $b>a\ge 3$ with $\gcd (a,b)=1$.

13 pages