Primes of the form $ax+by$

TL;DR

This paper investigates the distribution of primes within certain linear forms of two coprime positive integers, establishing lower bounds and conjecturing bounds related to the count of such primes up to a specific threshold.

Contribution

It provides a new lower bound on the number of primes of the form ax+by less than s(a,b) and proposes a conjecture relating this count to the total primes up to s(a,b).

Findings

Proves a lower bound for the number of primes in T(a,b) less than s(a,b).

Identifies specific cases where the count of such primes is zero or one.

Conjectures bounds relating the count of primes in T(a,b) to the total primes up to s(a,b).

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 $\gcd (a, b)=1$.