Sufficient conditions for solvability of linear Diophantine equations, and Frobenius numbers
Eteri Samsonadze
TL;DR
This work addresses the solvability of linear Diophantine equations in non-negative integers and the computation of Frobenius numbers for coprime coefficient sets. It develops a sufficient solvability condition based on the least common multiple $M$ of the coefficients and the remainder $r$ of $b$ modulo $M$, yielding concrete thresholds such as $b\ge (n-1)M$ or $b\ge nM-\sum a_i$. A counting-function framework $P(b)$ is used to derive necessary-and-sufficient solvability criteria (Theorem 2.1) and to connect solvability to Frobenius-number formulas for various structured families. A novel recurrent method reduces the problem stepwise and enables the computation of $g(a_1,\dots,a_n)$ for any $n\ge 3$, demonstrated by the explicit computation $g(6,8,11,13,15)=10$. Together, these results expand explicit Frobenius-number knowledge beyond small $n$ and provide constructive tools for assessing solvability in broad settings.
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 $n=5$.