Numerical Semigroups generated by Primes

Abstract

Let $p_1=2,p_2=3,p_3=5,\dots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that $\bullet$ $u_n\sim 3p_n$. Similarly, for the largest integer $f_n$ not contained in $S_n$, by computational evidence we suspect that $\bullet$ $f_n$ is an odd number for $n\ge 5$ and $\bullet$ $f_n\sim 3p_n$; further $\bullet$ $4p_n>f_{n+1}$ for $n\ge 1$. If $f_n$ is odd for large $n$, then $f_n\sim 3p_n$. In case $f_n\sim 3p_n$ every large even integer $x$ is the sum of two primes. If $4p_n>f_{n+1}$ for $n\ge 1$, then the Goldbach conjecture holds true. Further, Wilf's question in [12] has a positive answer for the semigroups $S_n$.

14 pages, 4 figures; added references for tables