Two-generator numerical semigroups and Fermat and Marsenne numbers

Shalom Eliahou (1) , Jorge Ramirez Alfonsin (2)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Given $g\in \N$, what is the number of numerical semigroups $S=\vs{a,b}$ in $\N$ of genus $|\N\setminus S|=g$? After settling the case $g=2^k$ for all $k$, we show that attempting to extend the result to $g=p^k$ for all odd primes $p$ is linked, quite surprisingly, to the factorization of Fermat and Mersenne numbers.

Mots clés: Semigroup; factorization; Fermat; Marsenne

Domaines: Combinatoire [math.CO]

Shalom Eliahou, Jorge Ramirez Alfonsin. Two-generator numerical semigroups and Fermat and Marsenne numbers. SIAM Journal on Discrete Mathematics, 2011, 25, pp.622-630. ⟨10.1137/100787283⟩. ⟨hal-00812743⟩