Comptes Rendus
no. 9-10
Number Theory
Primes in numerical semigroups
J.L. Ramírez Alfonsín1 ; M. Skałba2
1 UMI2924 - Jean- Christophe Yoccoz, CNRS-IMPA, Brazil and Univ. Montpellier, CNRS, Montpellier, France
2 Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1001-1004.

Let 0<a<b be two relatively prime integers and let 〈a,b〉 be the numerical semigroup generated by a and b with Frobenius number g(a,b)=ab-a-b. In this note, we prove that there exists a prime number p∈〈a,b〉 with p<g(a,b) when the product ab is sufficiently large. Two related conjectures are posed and discussed as well.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.104
Classification: 11D07, 11N13

J.L. Ramírez Alfonsín 1; M. Skałba 2

1 UMI2924 - Jean- Christophe Yoccoz, CNRS-IMPA, Brazil and Univ. Montpellier, CNRS, Montpellier, France
2 Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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J.L. Ramírez Alfonsín; M. Skałba. Primes in numerical semigroups. Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1001-1004. doi : 10.5802/crmath.104. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.104/

[1] Guadalupe Márquez-Campos; Jorge L. Ramírez Alfonsín; José M. Tornero Integral points in rational polygons: a numerical semigroup approach, Semigroup Forum, Volume 94 (2017) no. 1, pp. 123-138 | DOI | MR | Zbl

[2] Karl Prachar Primzahlverteilung, Die Grundlehren der Mathematischen Wissenschaften, 91, Springer, 1957 | MR | Zbl

[3] Jorge L. Ramírez Alfonsín The Diophantine Frobenius Problem, Oxford Lecture Series in Mathematics and its Applications, 30, Oxford University Press, 2005 | MR | Zbl

[4] Wacław Sierpiński Elementary Theory of Numbers, North-Holland Mathematical Library, 31, PWN-Polish Scientific Publisher; North-Holland, 1988 | MR | Zbl

[5] James Joseph Sylvester Question 7382, Mathematical Questions from Educational Times, Volume 41 (1884), p. 21

[6] Gérald Tenenbaum Introduction to Analytic and Probabilistic Number Theory, Graduate Studies in Mathematics, 163, American Mathematical Society, 2015 | MR | Zbl

[7] Arnold Walfisz Zur additiven Zahlentheorie. II, Math. Z., Volume 40 (1936) no. 1, pp. 592-607 | DOI | MR | Zbl

[8] Triantafyllos Xylouris Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen Progression, Bonner Mathematische Schriften, 404, University Bonn; Mathematisches Institut and Mathematisch-Naturwissenschaftliche Fakultät, 2011 | MR | Zbl

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