标题： 不属于数值半群的素数个数 显示英文标题

标题： The number of primes not in a numerical semigroup

摘要： 对于两个互质的正整数$a$和$b$，设$π^* (a, b)$是不能表示为$au+bv$的素数个数，其中$u$和$v$是非负整数。 显然，$π^* (a, b)\le π(ab-a-b)$，其中$π(x)$表示不超过$x$的素数个数。 本文中，我们证明了$π^* (a, b)\ge 0.04π(ab-a-b)$并提出以下猜想：$π^* (a, b)\ge \frac 12 π(ab-a-b)$。 该猜想已被证实对$1\le a\le 10$成立。 显示英文摘要

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