The Square Frobenius Number

TL;DR

This paper introduces and investigates a new variant of the Frobenius number, focusing on the largest square number not in a numerical semigroup, providing bounds, exact formulas, and conjectures for specific cases.

Contribution

It defines the square Frobenius number, derives bounds and exact formulas for certain semigroups, and proposes conjectures for broader cases.

Findings

Provided an upper bound for the square Frobenius number in arithmetic progression semigroups.

Derived exact formulas for semigroups generated by two numbers with specific differences.

Formulated conjectures for the square Frobenius number in general cases.

Abstract

Let $S=\left\langle s_1,\ldots,s_n\right\rangle$ be a numerical semigroup generated by the relatively prime positive integers $s_1,\ldots,s_n$. Let $k\geqslant 2$ be an integer. In this paper, we consider the following $k$-power variant of the Frobenius number of $S$ defined as $${}^{k\!}r\!\left(S\right):= \text{ the largest } k \text{-power integer not belonging to } S.$$In this paper, we investigate the case $k=2$. We give an upper bound for ${}^{2\!}r\!\left(S_A\right)$ for an infinite family of semigroups $S_A$ generated by {\em arithmetic progressions}. The latter turns out to be the exact value of ${}^{2\!}r\!\left(\left\langle s_1,s_2\right\rangle\right)$ under certain conditions. We present an exact formula for ${}^{2\!}r\!\left(\left\langle s_1,s_1+d \right\rangle\right)$ when $d=3,4$ and $5$, study ${}^{2\!}r\!\left(\left\langle s_1,s_1+1 \right\rangle\right)$ and ${}^{2\!}r\!\left(\left\langle s_1,s_1+2 \right\rangle\right)$ and put forward two relevant conjectures. We finally discuss some related questions.