On the number of numerical semigroups of prime power genus

TL;DR

This paper investigates the count of two-generator numerical semigroups of prime power genus, revealing that for certain cases, the count depends only on the prime's class modulo an explicit number, using gcd reductions and continued fractions.

Contribution

It establishes a dependence of the count of two-generator numerical semigroups of prime power genus on the prime's class modulo an explicit number, with detailed analysis for the case g=p^9.

Findings

n(p^k,2) depends only on p modulo M(k)

Explicit reduction of gcd expressions using continued fractions

Detailed case study for g=p^9 showing class dependence

Abstract

Given $g\ge 1$, the number $n(g)$ of numerical semigroups $S \subset \N$ of genus $|\N \setminus S|$ equal to $g$ is the subject of challenging conjectures of Bras-Amor\'os. In this paper, we focus on the counting function $n(g,2)$ of \textit{two-generator} numerical semigroups of genus $g$, which is known to also count certain special factorizations of $2g$. Further focusing on the case $g=p^k$ for any odd prime $p$ and $k \ge 1$, we show that $n(p^k,2)$ only depends on the class of $p$ modulo a certain explicit modulus $M(k)$. The main ingredient is a reduction of $\gcd(p^\alpha+1, 2p^\beta+1)$ to a simpler form, using the continued fraction of $\alpha/\beta$. We treat the case $k=9$ in detail and show explicitly how $n(p^9,2)$ depends on the class of $p$ mod $M(9)=3 \cdot 5 \cdot 11 \cdot 17 \cdot 43 \cdot 257$.