Numerical Semigroups generated by Primes

TL;DR

This paper investigates numerical semigroups generated by primes, providing asymptotic relations for their largest generators and Frobenius numbers, and explores implications for Goldbach's conjecture and Wilf's question.

Contribution

It introduces new asymptotic behaviors of generators and Frobenius numbers of prime-generated semigroups, linking them to classical conjectures.

Findings

$u_n o 3p_n$ as $n o ty$

Suspected $f_n$ is odd for large $n$ and $f_n o 3p_n$

If $f_n o 3p_n$, then every large even integer is sum of two primes

Abstract

Let $p_1=2, p_2=3, p_3=5, \ldots$ 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 $\bullet$ $u_n\sim3p_n$. 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 $n\geq5$ and $\bullet$ $f_n\sim3p_n$; further $\bullet$ $4p_n>f_{n+1}$ for $n\geq1$. If $f_n$ is odd for large $n$, then $f_n\sim3p_n$. In case $f_n\sim3p_n$ every large even integer $x$ is the sum of two primes. If $4p_n>f_{n+1}$ for $n\geq1$, then the Goldbach conjecture holds true. Further, Wilf's question in [12] has a positive answer for the semigroups $S_n$.