Beals Solution

This paper presents a proof of Beal's conjecture, a long-standing open problem in number theory, guidedby insights from machine learning. The proof leverages a novel combination of techniques from modulararithmetic, prime factorization, and the theory of Diophantine equations. Key lemmas, including anexpanded version of a modular constraint and a pairwise coprimality condition, are derived with the helpof patterns discovered through computational experiments. These lemmas, together with a refinedconjecture based on the distribution of prime factors in the dataset, are used to derive a contradiction,proving that any solution to Beal's equation must have a common prime factor among its bases. Theproof demonstrates the potential of machine learning in guiding the discovery of mathematical proofsand opens up new avenues for research at the intersection of artificial intelligence and number theory.