On the determination of $p$-Frobenius and related numbers using the $p$-Ap\'ery set

TL;DR

This paper introduces explicit formulas for the generalized $p$-Frobenius number and related values using the $p$-Apéry set, extending classical Frobenius problem solutions systematically.

Contribution

It provides a systematic method to compute the $p$-Frobenius number and related quantities via new formulas involving the $p$-Apéry set, generalizing classical results.

Findings

Explicit formulas for $p$-Frobenius number and related values.

Systematic computation method for generalized Frobenius problems.

Generalization of weighted sums using $p$-Apéry set.

Abstract

In this paper, we give convenient formulas in order to obtain explicit expressions of a generalized Frobenius number called the $p$-Frobenius number as well as its related values. Here, for a non-negative integer $p$, the $p$-Frobenius number is the largest integer whose number of solutions of the linear diophantine equation in terms of positive integers $a_1,a_2,\dots,a_k$ with $\gcd(a_1,a_2,\dots,a_k)=1$ is at most $p$. When $p=0$, the problem is reduced to the famous and classical linear Diophantine problem of Frobenius. $0$-Frobenius number is the classical Frobenius number. Our formula is not only a natural extension of the existing classical formulas, but also has the great advantage that the explicit expressions of values such as the $p$-Frobenius and related numbers can be obtained systematically. The concept and formula of the weighted sum has been given recently. We also give a $p$-generalized formula for such weighted sums. The central role is the $p$-Ap\'ery set, which is a generalization of the classical Ap\'ery set.