High Degree Diophantine Equation c^q=a^p+b^p

TL;DR

This paper explores algebraic structures in integer modules to analyze and solve high-degree Diophantine equations, introducing a novel approach that leverages module theory for number theory problems.

Contribution

It presents a new method for studying integer functions in modules and applies it to solve complex high-degree Diophantine equations.

Findings

Proves a result relating unequal logarithms in modules

Develops a new algebraic approach to Diophantine equations

Demonstrates application to specific high-degree equations

Abstract

The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction, a result is proven that two finite numbers is with unequal logarithms in a corresponding module, and is applied to solving a kind of high degree diophantine equation.