On the number of numerical semigroups of prime power genus (original) (raw)

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

Semigroup Forum, 2012

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 gcd(p α + 1, 2p β + 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 mod M (9) = 3 • 5 • 11 • 17 • 43 • 257.

On the number of numerical . . . of prime power genus

2014

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 gcd(p α + 1, 2p β + 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 mod M (9) = 3 • 5 • 11 • 17 • 43 • 257.

Two-Generator Numerical Semigroups and Fermat and Mersenne Numbers

SIAM Journal on Discrete Mathematics, 2011

Given g ∈ N, what is the number of numerical semigroups S = a, b in N of genus |N \ S| = g? After settling the case g = 2 k for all k, we show that attempting to extend the result to g = p k for all odd primes p is linked, quite surprisingly, to the factorization of Fermat and Mersenne numbers.

Distribution of genus among numerical semigroups with fixed Frobenius number

Semigroup Forum

A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number f and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number f have genus close to \frac{3f}{4}3f4.WedenotethenumberofnumericalsemigroupsofFrobeniusnumberfbyN(f).WhileN(f)isnotmonotonicweprovethat3 f 4 . We denote the number of numerical semigroups of Frobenius number f by N(f). While N(f) is not monotonic we prove that3f4.WedenotethenumberofnumericalsemigroupsofFrobeniusnumberfbyN(f).WhileN(f)isnotmonotonicweprovethatN(f)

On the genus of a quotient of a numerical semigroup

Semigroup Forum, 2019

We find a relation between the genus of a quotient of a numerical semigroup S and the genus of S itself. We use this identity to compute the genus of a quotient of S when S has embedding dimension 2. We also exhibit identities relating the Frobenius numbers and the genus of quotients of numerical semigroups that are generated by certain types of arithmetic progressions.

The genus, Frobenius number and pseudo-Frobenius numbers of numerical semigroups of type 2

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2016

We study some questions on numerical semigroups of type 2. On the one hand, we investigate the relation between the genus and the Frobenius number. On the other hand, for two fixed positive integersg1,g2, we give necessary and sufficient conditions in order to have a numerical semigroupSsuch that {g1,g2} is the set of its pseudo-Frobenius numbers and, moreover, we explicitly build families of such numerical semigroups.

On Numerical Semigroups with Almost-Maximal Genus

2020

A numerical semigroup is a cofinite subset of N0, containing 0, that is closed under addition. Its genus is the number of nonnegative integers that are missing. A numerical set is a similar object, not necessarily closed under addition. If T is a numerical set, then A(T)={n in N0 : n+T is a subset of T} is a numerical semigroup. Recently a paper appeared counting the number of numerical sets T where A(T) is a numerical semigroup of maximal genus. We count the number of numerical sets T where A(T) is a numerical semigroup of almost-maximal genus, i.e. genus one smaller than maximal.

Modular retractions of numerical semigroups

Semigroup Forum, 2013

Let S be a numerical semigroup, let m be a nonzero element of S, and let a be a nonnegative integer. We denote R(S, a, m) = {s − as mod m | s ∈ S} (where as mod m is the remainder of the division of as by m). In this paper we characterize the pairs (a, m) such that R(S, a, m) is a numerical semigroup. In this way, if we have a pair (a, m) with such characteristics, then we can reduce the problem of computing the genus of S = n 1 ,. .. , n p to computing the genus of a "smaller" numerical semigroup n 1 − an 1 mod m,. .. , n p − an p mod m. This reduction is also useful for estimating other important invariants of S such as the Frobenius number and the type. Keywords Modular retractions • Numerical semigroups • Apéry sets • Modular translations 1 Introduction Let N be the set of nonnegative integers. A numerical semigroup is a subset S of N such that it is closed under addition, 0 ∈ S and N \ S is finite.