Generalized Frobenius Number of Three Variables

TL;DR

This paper derives a formula for the generalized Frobenius number for three positive integers under specific conditions, extending classical results to a broader context involving multiple representations.

Contribution

It provides a new explicit formula for the generalized Frobenius number of three integers, which was previously unknown in certain cases.

Findings

Derived a formula for the generalized Frobenius number for three integers

Extended classical Frobenius number results to multiple representations

Applicable under specific conditions for the integers involved

Abstract

For $ k \geq 2 $, we let $ A = (a_{1}, a_{2}, \ldots, a_{k}) $ be a $k$-tuple of positive integers with $\gcd(a_{1}, a_2, \ldots, a_k) =1$ and, for a non-negative integer $s$, the generalized Frobenius number of $A$, $g(A;s) = g(a_1, a_2, \ldots, a_k;s)$, the largest integer that has at most $s$ representations in terms of $a_1, a_2, \ldots, a_k$ with non-negative integer coefficients. In this article, we give a formula for the generalized Frobenius number of three positive integers $(a_1,a_2,a_3)$ with certain conditions.