Problems Regarding a Pair of Diophantine Equations

TL;DR

This paper reviews recent and new findings on the solutions of specific Diophantine equations involving coprime positive integers, highlighting their unique solutions and proposing future research directions.

Contribution

It summarizes recent results and introduces new insights into the solutions of two particular Diophantine equations, and outlines open problems for future study.

Findings

Exactly one of the two equations has a nonnegative integer solution for coprime positive integers.

The solution to the equation, when it exists, is unique.

The paper discusses open problems and future research directions in this area.

Abstract

For two relatively prime positive integers $a, b\in \mathbb{N}$, it is known that exactly one of the two Diophantine equations $$ax + by \ =\ \frac{(a-1)(b-1)}{2}\ \mbox{ and }\ 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}$$ has a nonnegative integral solution $(x, y)$. Furthermore, the solution is unique. In this note, we summarize recent results and some new ones on the solution of the two equations and provide an overview of problems for future investigation, some of which were presented at the 2025 International Conference on Class Groups of Number Fields and Related Topics.