Numerical Semigroups generated by Primes
Michael Hellus, Anton Rechenauer, Rolf Waldi

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 be the consecutive prime numbers, the numerical semigroup generated by the primes not less than and the largest irredundant generator of . We will show, that . Similarly, for the largest integer not contained in , by computational evidence we suspect that is an odd number for and ; further for . If is odd for large , then . In case every large even integer is the sum of two primes. If for , then the Goldbach conjecture holds true. Further, Wilf's question in [12] has a positive answer for the semigroups .
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