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Semigroup Forum

, Volume 87, Issue 1, pp 171–186 | Cite as

On the number of numerical semigroups 〈a,b〉 of prime power genus

  • Shalom Eliahou
  • Jorge Ramírez Alfonsín
RESEARCH ARTICLE

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 2g. 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 \(\operatorname{gcd}(p^{\alpha}+1, 2p^{\beta}+1)\) 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 \(p \operatorname{mod} M(9)=3 \cdot5 \cdot11 \cdot17 \cdot43 \cdot257\).

Keywords

Gap number Sylvester’s theorem Special factorizations Euclidean algorithm Continued fractions RSA 

References

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Univ. Lille Nord de FranceLilleFrance
  2. 2.ULCO, CNRS FR 2956LMPA J. LiouvilleCalaisFrance
  3. 3.Institut de Mathématiques et de Modélisation de Montpellier, UMRS 5149, CNRSUniversité Montpellier 2MontpellierFrance

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