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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Bibliographic Details
Published in:Semigroup forum Vol. 87; no. 1; pp. 171 - 186
Main Authors: Eliahou, Shalom, Ramírez Alfonsín, Jorge
Format: Journal Article
Language:English
Published: Boston Springer US 01.08.2013
Subjects:
Algebra
Mathematics
Mathematics and Statistics
Research Article
Special factorizations
Sylvester’s theorem
RSA
Gap number
Euclidean algorithm
Continued fractions
ISSN:0037-1912, 1432-2137
Online Access:Get full text
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Summary: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 .
ISSN:0037-1912
1432-2137
DOI:10.1007/s00233-012-9457-4