The number of primes not in a numerical semigroup

Abstract

For two coprime positive integers $a$ and $b$,let ${\pi }^{\ast }(a,b)$ be the number of primes that cannot be represented as $au+bv$, where $u$ and $v$ are nonnegative integers. It is clear that ${\pi }^{\ast }(a,b)\le \pi (ab-a-b)$, where $\pi (x)$ denotes the number of primes not exceeding $x$. In this paper, we prove that ${\pi }^{\ast }(a,b)\ge 0.04\pi (ab-a-b)$ and pose following conjecture: ${\pi }^{\ast }(a,b)\ge \frac{1}{2}\pi (ab-a-b)$. This conjecture is confirmed for $1\le a\le 10$.

12 pages