The Square Frobenius Number - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2020

The Square Frobenius Number

Résumé

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 infinity 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.
Fichier principal
Vignette du fichier
SquareFrobeniusNumber.pdf (345.9 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-02879706 , version 1 (24-06-2020)
hal-02879706 , version 2 (02-11-2020)
hal-02879706 , version 3 (29-04-2022)
hal-02879706 , version 4 (23-05-2022)

Identifiants

Citer

Jonathan Chappelon, Jorge Luis Ramírez Alfonsín. The Square Frobenius Number. 2020. ⟨hal-02879706v2⟩
218 Consultations
105 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More