# Note for the Beal's Conjecture

### EasyChair Preprint 13256, version 6

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6 pagesDate: May 17, 2024

### 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.