Solving Linear Diophantine Equations Using Generalized Chinese Remainder Theorem

Abstract

This project presents a new framework for solving linear Diophantine equations by using Generalized Chinese Remainder Theorem (GCRT). Standard approaches are often computationally heavy and require deep understanding of advanced number theory. To address this, we build on Ming Xiong’s “minimum principles”- a function based technique for simplification – to develop an algorithm that converts a system of linear Diophantine equations into a GCRT problem.

 Crucially, our method handles these systems regardless of whether the moduli are coprime. By combining Xiong’s concepts of minimum-principle-based transformations with the GCRT framework, our algorithm simplifies the solution process, making it both more accessible and computationally efficient. Ultimately this study provides a more generalized and practical tool for finding integer solutions, addressing a significant gap in the existing literature. The proposed method demonstrates how fundamental mathematical principles can be applied in new contexts to simplify complex problems, offering a clear and straightforward alternatives to conventional methods.