On the least positive solution to a proportionally modular Diophantine inequality

Abstract

Given three positive integers $a,b,c$, a proportionally modular Diophantine inequality is an expression of the form $ax\mathrm{mod}b\le cx$. 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 García-Sánchez.