Note for the Beal's Conjecture

Abstract

This work explores two famous conjectures in number theory: Fermat's Last Theorem and Beal's Conjecture. Fermat's Last Theorem, posed by Pierre de Fermat in the 17th century, states that there are no positive integer solutions for the equation $a^{n} + b^{n} = c^{n}$, where $n$ is greater than $2$. This theorem remained unproven for centuries until Andrew Wiles published a proof in 1994. Beal's Conjecture, formulated in 1997 by Andrew Beal, generalizes Fermat's Last Theorem. It states that for positive integers $A$, $B$, $C$, $x$, $y$, and $z$, if $A^{x} + B^{y} = C^{z}$ (where $x$, $y$, and $z$ are all greater than $2$), then $A$, $B$, and $C$ must share a common prime factor. Beal's Conjecture remains unproven, and a significant prize is offered for a solution. This paper provides a concise introduction to both conjectures, highlighting their connection and presenting a short proof of the Beal's Conjecture.