The number of primes not in a numerical semigroup

TL;DR

This paper investigates the count of primes that cannot be expressed as a linear combination of two coprime positive integers, establishing bounds and conjecturing a proportional relationship.

Contribution

It proves a lower bound for the number of such primes and formulates a conjecture relating this count to the total primes up to a certain limit.

Findings

Established that (a, b) \u2265 0.04 (ab - a - b)

Conjectured (a, b) \u2265 0.5 (ab - a - b)

Confirmed the conjecture for 1                     

Abstract

For two coprime positive integers $a$ and $b$,let $\pi^* (a, b)$ be the number of primes that cannot be represented as $au+bv$, where $u$ and $v$ are nonnegative integers. It is clear that $\pi^* (a, b)\le \pi (ab-a-b)$, where $\pi (x)$ denotes the number of primes not exceeding $x$. In this paper, we prove that $\pi^* (a, b)\ge 0.04\pi (ab-a-b)$ and pose following conjecture: $\pi^* (a, b)\ge \frac 12 \pi (ab-a-b)$. This conjecture is confirmed for $1\le a\le 10$.