ABC Conjecture and Fermat's Last Theorem: A Unified Diophantine Framework

This paper explores the profound and often intertwined relationship between two monumental concepts in number theory: Fermat's Last Theorem (FLT) and the ABC Conjecture. While FLT, famously proven by Andrew Wiles, stands as a testament to the power of highly specialized techniques in algebraic geometry and modular forms, the ABC Conjecture offers a strikingly general framework that has far-reaching implications for a vast class of Diophantine equations. We investigate how the ABC Conjecture, despite remaining unproven, provides an elegant and concise heuristic argument for FLT, particularly for sufficiently large exponents, and discuss its potential to unify our understanding of integer solutions to equations of the form $A+B=C$. By examining the core statements and consequences of both FLT and the ABC Conjecture, we aim to delineate a unified Diophantine framework that highlights the predictive power and structural elegance of the latter, contrasting it with the specificity of Wiles' celebrated proof. The discussion extends to the broader implications for Diophantine analysis, considering both the advantages and inherent limitations of such a generalized approach.