On the number of numerical semigroups 〈a,b〉 of prime power genus
Given g ≥1, the number n ( g ) of numerical semigroups S ⊂ℕ of genus |ℕ∖ S | equal to g is the subject of challenging conjectures of Bras-Amorós. In this paper, we focus on the counting function n ( g ,2) of two-generator numerical semigroups of genus g , which is known to also count certain special...
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| Published in: | Semigroup forum Vol. 87; no. 1; pp. 171 - 186 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Boston
Springer US
01.08.2013
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| Subjects: | |
| ISSN: | 0037-1912, 1432-2137 |
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| Abstract | Given
g
≥1, the number
n
(
g
) of numerical semigroups
S
⊂ℕ of genus |ℕ∖
S
| equal to
g
is the subject of challenging conjectures of Bras-Amorós. In this paper, we focus on the counting function
n
(
g
,2) of
two-generator
numerical semigroups of genus
g
, which is known to also count certain special factorizations of 2
g
. Further focusing on the case
g
=
p
k
for any odd prime
p
and
k
≥1, we show that
n
(
p
k
,2) only depends on the class of
p
modulo a certain explicit modulus
M
(
k
). The main ingredient is a reduction of
to a simpler form, using the continued fraction of
α
/
β
. We treat the case
k
=9 in detail and show explicitly how
n
(
p
9
,2) depends on the class of
. |
|---|---|
| AbstractList | Given
g
≥1, the number
n
(
g
) of numerical semigroups
S
⊂ℕ of genus |ℕ∖
S
| equal to
g
is the subject of challenging conjectures of Bras-Amorós. In this paper, we focus on the counting function
n
(
g
,2) of
two-generator
numerical semigroups of genus
g
, which is known to also count certain special factorizations of 2
g
. Further focusing on the case
g
=
p
k
for any odd prime
p
and
k
≥1, we show that
n
(
p
k
,2) only depends on the class of
p
modulo a certain explicit modulus
M
(
k
). The main ingredient is a reduction of
to a simpler form, using the continued fraction of
α
/
β
. We treat the case
k
=9 in detail and show explicitly how
n
(
p
9
,2) depends on the class of
. |
| Author | Ramírez Alfonsín, Jorge Eliahou, Shalom |
| Author_xml | – sequence: 1 givenname: Shalom surname: Eliahou fullname: Eliahou, Shalom organization: Univ. Lille Nord de France, ULCO, CNRS FR 2956, LMPA J. Liouville – sequence: 2 givenname: Jorge surname: Ramírez Alfonsín fullname: Ramírez Alfonsín, Jorge email: jramirez@math.univ-montp2.fr organization: Institut de Mathématiques et de Modélisation de Montpellier, UMRS 5149, CNRS, Université Montpellier 2 |
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| Issue | 1 |
| Keywords | Special factorizations Sylvester’s theorem RSA Gap number Euclidean algorithm Continued fractions |
| Language | English |
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| References | Bras-Amorós (CR1) 2008; 76 Sylvester (CR6) 1882; 5 Bras-Amorós (CR2) 2009; 213 Eliahou, Ramírez Alfonsín (CR3) 2011; 25 Ramírez Alfonsín (CR4) 2005 Rosales, García-Sánchez (CR5) 2009 J.J. Sylvester (9457_CR6) 1882; 5 M. Bras-Amorós (9457_CR1) 2008; 76 M. Bras-Amorós (9457_CR2) 2009; 213 S. Eliahou (9457_CR3) 2011; 25 J.C. Rosales (9457_CR5) 2009 J.L. Ramírez Alfonsín (9457_CR4) 2005 |
| References_xml | – year: 2005 ident: CR4 publication-title: The Diophantine Frobenius Problem doi: 10.1093/acprof:oso/9780198568209.001.0001 – year: 2009 ident: CR5 publication-title: Numerical Semigroups doi: 10.1007/978-1-4419-0160-6 – volume: 213 start-page: 997 year: 2009 end-page: 1001 ident: CR2 article-title: Bounds on the number of numerical semigroups of a given genus publication-title: J. Pure Appl. Algebra doi: 10.1016/j.jpaa.2008.11.012 – volume: 5 start-page: 119 year: 1882 end-page: 136 ident: CR6 article-title: On subinvariants, i.e. semi-invariants to binary quantities of an unlimited order publication-title: Am. J. Math. – volume: 25 start-page: 622 year: 2011 end-page: 630 ident: CR3 article-title: Two-generator numerical semigroups and Fermat and Mersenne numbers publication-title: SIAM J. Discrete Math. doi: 10.1137/100787283 – volume: 76 start-page: 379 year: 2008 end-page: 384 ident: CR1 article-title: Fibonacci-like behavior of the number of numerical semigroups of a given genus publication-title: Semigroup Forum doi: 10.1007/s00233-007-9014-8 – volume: 76 start-page: 379 year: 2008 ident: 9457_CR1 publication-title: Semigroup Forum doi: 10.1007/s00233-007-9014-8 – volume-title: Numerical Semigroups year: 2009 ident: 9457_CR5 doi: 10.1007/978-1-4419-0160-6 – volume: 5 start-page: 119 year: 1882 ident: 9457_CR6 publication-title: Am. J. Math. – volume: 25 start-page: 622 year: 2011 ident: 9457_CR3 publication-title: SIAM J. Discrete Math. doi: 10.1137/100787283 – volume: 213 start-page: 997 year: 2009 ident: 9457_CR2 publication-title: J. Pure Appl. Algebra doi: 10.1016/j.jpaa.2008.11.012 – volume-title: The Diophantine Frobenius Problem year: 2005 ident: 9457_CR4 doi: 10.1093/acprof:oso/9780198568209.001.0001 |
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| Snippet | Given
g
≥1, the number
n
(
g
) of numerical semigroups
S
⊂ℕ of genus |ℕ∖
S
| equal to
g
is the subject of challenging conjectures of Bras-Amorós. In this... |
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| StartPage | 171 |
| SubjectTerms | Algebra Mathematics Mathematics and Statistics Research Article |
| Title | On the number of numerical semigroups 〈a,b〉 of prime power genus |
| URI | https://link.springer.com/article/10.1007/s00233-012-9457-4 |
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