The Square Frobenius Number

Abstract

Let $S=⟨s_1,\dots ,s_n⟩$ be a numerical semigroup generated by the relatively prime positive integers $s_1,\dots ,s_n$. Let $k⩾2$ 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 ${}^{2}r\left(⟨s_1,s_2⟩\right)$ under certain conditions. We present an exact formula for ${}^{2}r\left(⟨s_1,s_1+d⟩\right)$ when $d=3,4$ and $5$, study ${}^{2}r\left(⟨s_1,s_1+1⟩\right)$ and ${}^{2}r\left(⟨s_1,s_1+2⟩\right)$ and put forward two relevant conjectures. We finally discuss some related questions.