Sufficient conditions for solvability of linear Diophantine equations, and Frobenius numbers

TL;DR

This paper establishes sufficient conditions for the solvability of linear Diophantine equations in non-negative integers, provides explicit formulas for Frobenius numbers in certain cases, and introduces a new recursive method for analyzing these problems.

Contribution

It introduces a new recursive approach to determine Frobenius numbers for any number of variables, extending previous methods and providing explicit formulas for specific cases.

Findings

Sufficient conditions for solvability of linear Diophantine equations in non-negative integers.

Explicit formulas for Frobenius numbers in particular cases.

A new recursive method for computing Frobenius numbers for any number of variables.

Abstract

The sufficient conditions for solvability of a linear Diophantine equation $\sum_{i=1}^{n}a_ix_i=b$ (with $a_1,a_2,...,a_n\in \mathbb{N}$) in non-negative integers $x_1,x_2,...,x_n$ are given. The explicit formulas are given for Frobenius numbers $g(a_1,a_2,...,a_n)$, for some particular cases,. Besides, a new recurrent method of studying the problem of solvability of a linear Diophantine equation in non-negative integers is proposed. This recurrent method is used for the problem of finding Frobenius numbers $g(a_1,a_2,...,a_n)$ for any $n\geq 3$; the example is given for the case $n=5$.