The Relationship Between the Exponential Diophantine Equation of the Form a^x + b^y = z^2 and Elliptic Curves in modulo 4 and modulo 6

Abstract

This thesis investigates the relationship between exponential Diophantine equations of the form a^x+b^y=z^2 and elliptic curves, with particular emphasis on transformations modulo 4 and modulo 6. The work presents a clear methodological framework for converting these equations into elliptic curve forms, allowing the application of algebraic and computational tools. It demonstrates how modular considerations simplify the structure of the problem and lead to deeper insights into the existence of integer solutions. The integration of theoretical analysis with computational techniques, including SageMath implementations, strengthens the overall approach. The exposition is systematic and accessible, guiding the reader from foundational concepts to more advanced results. Overall, the thesis provides a meaningful contribution to the study of Diophantine equations and highlights the power of elliptic curve methods in number theory.