The Square Frobenius Number

Jonathan Chappelon (1) , Jorge Luis Ramírez Alfonsín (1, 2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 IMPA Jean-Christophe Yoccoz

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.

Mots clés: Numerical semigroups; Frobenius number; Perfect square integer

Domaines: Théorie des nombres [math.NT]

Jonathan Chappelon, Jorge Luis Ramírez Alfonsín. The Square Frobenius Number. Semigroup Forum, 2022, 105, pp.149-171. ⟨10.1007/s00233-022-10289-z⟩. ⟨hal-02879706v4⟩