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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| Published in | SIAM journal on discrete mathematics Vol. 25; pp. 622 - 630 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Society for Industrial and Applied Mathematics
04.11.2011
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| Subjects | |
| Online Access | Get full text |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Ramirez Alfonsin, Jorge Eliahou, Shalom |
| Author_xml | – sequence: 1 givenname: Shalom orcidid: 0000-0002-2305-6205 surname: Eliahou fullname: Eliahou, Shalom organization: Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville – sequence: 2 givenname: Jorge surname: Ramirez Alfonsin fullname: Ramirez Alfonsin, Jorge organization: Institut de Mathématiques et de Modélisation de Montpellier |
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| Keywords | Marsenne Semigroup Fermat factorization |
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| Title | Two-generator numerical semigroups and Fermat and Marsenne numbers |
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