The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions

Komatsu, Takao; Pita-Ruiz, Claudio

doi:10.3390/axioms12020098

The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions

Journal

Axioms

ISSN

2075-1680

Date Issued

2023

Author(s)

Komatsu, Takao

Pita-Ruiz, Claudio

Facultad de Ingeniería - CampCM

Type

text::journal::journal article

DOI

10.3390/axioms12020098

URL

https://scripta.up.edu.mx/handle/20.500.12552/3672

Abstract

In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For (Formula presented.), the problem is reduced to the famous linear Diophantine problem of Frobenius, the largest integer of which is called the Frobenius number. We also give a closed formula for the number of non-negative integers (p-genus), such that linear equations have at most p representations. Extensions to the Jacobsthal polynomial and the Jacobsthal–Lucas polynomial give more general formulas that include the familiar Fibonacci and Lucas numbers. A basic problem with the Fibonacci triplet was dealt by Marin, Ramírez Alfonsín and M. P. Revuelta for (Formula presented.) and by Komatsu and Ying for the general non-negative integer p. © 2023 by the authors.