Primes in numerical semigroups

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 relate...

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Bibliographic Details
Published inComptes rendus. Mathématique Vol. 358; no. 9-10; pp. 1001 - 1004
Main Authors Ramírez Alfonsín, J.L., Skałba, M.
Format Journal Article
LanguageEnglish
Published Académie des sciences (Paris) 05.01.2021
Académie des sciences
Subjects
Mathematics
Number Theory
Numerical semigroups
Frobenius number
Primes
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Summary: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.
ISSN:1778-3569
1631-073X
1778-3569
DOI:10.5802/crmath.104