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The extended Frobenius problem for Lucas series incremented by a Lucas number

Aureliano M. Robles-Pérez, José Carlos Rosales

TL;DR

This work addresses the extended Frobenius problem for Lucas-number-based numerical semigroups, focusing on two families: $S(a)=\langle l_a, l_a+l_0,\dots, l_a+l_{a-1}\rangle$ and $T(a)=\langle l_a+l_0,\dots, l_a+l_a\rangle$. Leveraging the Zeckendorf-Lucas decomposition and Apéry sets, the authors obtain explicit generators and closed formulas for the Frobenius number and genus, notably $F(S(a))=\left\lceil \frac{a-1}{2} \right\rceil l_a-1$ and $g(S(a))=\frac{a}{5}(l_a+l_{a-2})$, while establishing Wilf's conjecture for these families. They also relate the two families via $S(a)=T(a)\cup\{l_a,2l_a+1\}$, with $T(a)$ offering a parallel set of results and satisfying Wilf's conjecture. Overall, the paper extends the extended Frobenius problem framework to Lucas-based semigroups and provides computable formulas for key invariants.

Abstract

We study the extended Frobenius problem for sequences of the form $\{l_a\}\cup\{l_a+l_n\}_{n\in\mathbb{N}}$ and $\{l_a+l_n\}_{n\in\mathbb{N}}$, where $\{l_n\}_{n\in\mathbb{N}}$ is the Lucas series and $l_a$ is a Lucas number. As a consequence, we show that the families of numerical semigroups associated to both sequences satisfy the Wilf's conjecture.

The extended Frobenius problem for Lucas series incremented by a Lucas number

TL;DR

This work addresses the extended Frobenius problem for Lucas-number-based numerical semigroups, focusing on two families: and . Leveraging the Zeckendorf-Lucas decomposition and Apéry sets, the authors obtain explicit generators and closed formulas for the Frobenius number and genus, notably and , while establishing Wilf's conjecture for these families. They also relate the two families via , with offering a parallel set of results and satisfying Wilf's conjecture. Overall, the paper extends the extended Frobenius problem framework to Lucas-based semigroups and provides computable formulas for key invariants.

Abstract

We study the extended Frobenius problem for sequences of the form and , where is the Lucas series and is a Lucas number. As a consequence, we show that the families of numerical semigroups associated to both sequences satisfy the Wilf's conjecture.
Paper Structure (7 sections, 29 theorems, 15 equations)

This paper contains 7 sections, 29 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: numerical semigroups
  3. The minimal system of generators of $S(a)$
  4. The Apéry set of $S(a)$
  5. The Frobenius number of $S(a)$
  6. The genus of $S(a)$
  7. Another family of numerical semigroups related to the Lucas series

Key Result

Proposition 2.1

Let $S$ be a numerical semigroup and $n\in S\setminus\{0\}$. Then the cardinality of $\mathrm{Ap}(S,n)$ is $n$. Moreover, where $w(i)$ is the least element of $S$ congruent with $i$ modulo $n$.

Theorems & Definitions (53)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Example 3.4
  • Lemma 4.1
  • Remark 4.2
  • ...and 43 more