On the least positive solution to a proportionally modular Diophantine inequality

TL;DR

This paper develops a recursive formula and an algorithm to find the least positive solution to a specific type of Diophantine inequality involving modular arithmetic, with applications to existing mathematical questions.

Contribution

It introduces a novel recursive approach and algorithm for solving proportionally modular Diophantine inequalities, extending previous work in the field.

Findings

Derived a recursive formula for the least solution.

Developed an algorithm based on the formula.

Applied results to a question by Rosales and García-Sánchez.

Abstract

Given three positive integers $a,b,c$, a proportionally modular Diophantine inequality is an expression of the form $ax \mod{b} \le cx$. Our aim is to give a recursive formula for the least solution to such an inequality. We then use the formula to derive an algorithm. Finally, we apply our results to a question of Rosales and Garc\'ia-S\'anchez.