Some Bounds for Number of Solutions to $ax + by + cz = n$ and their Applications

TL;DR

This paper derives bounds for the number of solutions to the linear Diophantine equation in three variables and applies these bounds to solve special cases of a Frobenius coin problem generalization and to disprove a recent conjecture.

Contribution

It introduces simple bounds for the solution count of $ax+by+cz=n$ and applies them to specific problems and conjectures in number theory.

Findings

Established bounds for the number of solutions.

Solved special cases of a generalized Frobenius coin problem.

Disproved a recent conjecture on solution structure.

Abstract

In a recent work, the present author developed an efficient method to find the number of solutions of $ax+by+cz=n$ in non-negative integer triples $(x,y,z)$ where $a,b,c$ and $n$ are given natural numbers. In this note, we use that formula to obtain some simple looking bounds for the number of solutions of $ax+by+cz=n$. Using these bounds, we solve some special cases of a problem related to the generalization of Frobenius coin problem in three variables. Moreover, we use these bounds to disprove a recent conjecture of He, Shiue and Venkat regarding the solution structure of $ax+by+cz=n$.