A Note on the Beal Conjecture

Around 1637, Pierre de Fermat famously wrote in the margin of a book that he had a proof showing the equation $a^n + b^n = c^n$ has no positive integer solutions for exponents $n$ greater than 2. This statement, known as Fermat's Last Theorem, remained unproven for over three centuries despite efforts by countless mathematicians. In 1994, Andrew Wiles finally provided a rigorous proof using advanced techniques from elliptic curves and modular forms—methods far beyond those available in Fermat's era. Wiles was awarded the Abel Prize in 2016, with the citation describing his work as a ``stunning advance'' in mathematics. The Beal conjecture, formulated in 1993, generalizes Fermat's Last Theorem. It states that if $A^{x} + B^{y} = C^{z}$ holds for positive integers $A, B, C, x, y, z$ with $x, y, z > 2$, then $A$, $B$, and $C$ must share a common prime factor. In this paper, we prove the Beal conjecture using elementary methods involving parametrization of quadratic Diophantine equations, divisibility properties, and congruence relations. Our approach potentially offers a solution closer in spirit to the mathematical tools available in Fermat's time.