On the number of integer non-negative solutions of a linear Diophantine equation

TL;DR

This paper presents a novel approach to count non-negative integer solutions of linear Diophantine equations using properties of the Kronecker function and combinatorics, avoiding traditional number series methods.

Contribution

It derives formulas and recurrence relations for calculating the solution count, including explicit formulas for coprime coefficients, expanding the computational tools for such problems.

Findings

Derived formulas express P(b) via P(r), P(r+M), ...

Recurrence relations for calculating P(b)

Explicit formula for coprime coefficients case

Abstract

We deal with the problem to find the number $P(b)$ of integer non-negative solutions of an equation $\sum_{i=1}^{n} a_i x_i=b$, where $a_1,a_2,...,a_n$ are natural numbers and $b$ is a non-negative integer. As different from the traditional methods of investigation of the function $P(b)$, in our study we do not employ the techniques of number series theory, but use in the main the properties of the Kronecker function and the elements of combinatorics. The formula is derived to express $P(b)$, for an integer non-negative $b$, via $P(r)$, $P(r+M)$,..., $P(r+(s-1)M)$ when $s\neq 0$, where $s=\left[ n-\dfrac{\sum_{i=1}^{n}a_i+r}{M} \right]$ and takes quite small values in some particular cases; $M$ is the least common multiple of the numbers $a_1,a_2,\ldots,a_n$, and $r$ is the remainder of $b$ modulo $M$. Also, the recurrent formulas are derived to calculate $P(b)$, for any non-negative integer $b$, which, in particular, are used in finding $P(r)$, $P(r+M)$,..., $P(r+(s-1)M)$. For the case where $s=0$ and $a_1$, $a_2$,..., $a_n$ are coprime, the explicit formula $P(b)=\dfrac{M^{n-1}}{a_1a_2\ldots a_n}C^{n-1}_{\left[\dfrac{b}{M}\right]+n-1}$ is given. To illustrate the proposed method, examples of finding the function $P(b)$ for linear Diophantine equations with $2,3,7$ and $n$ variables are given.