Two-generator numerical semigroups and Fermat and Marsenne numbers

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...

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Bibliographic Details
Published inSIAM journal on discrete mathematics Vol. 25; pp. 622 - 630
Main Authors Eliahou, Shalom, Ramirez Alfonsin, Jorge
Format Journal Article
LanguageEnglish
Published Society for Industrial and Applied Mathematics 04.11.2011
Subjects
Combinatorics
Mathematics
Marsenne
Semigroup
Fermat
factorization
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Summary: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.
ISSN:0895-4801
DOI:10.1137/100787283