Proof for the Beal Conjecture and a New Proof for Fermat's Last Theorem

Abstract

The Beal Conjecture was formulated in 1997 and presented as a generalization of Fermat's Last Theorem, within the number theory´s field. It states that, for X, Y, Z, n₁, n₂ and n₃ positive integers with n₁, n₂, n₃ > 2, if Xⁿ¹ +Yⁿ²=Zⁿ³ then X, Y, Z must have a common prime factor. This article aims to present developments on Beal Conjecture, obtained from the correspondences between the real solutions of the equations in the forms A² + B² = C² (here simply refereed as Pythagoras´ equation), δⁿ + γⁿ=αⁿ (here simply refereed as Fermat´s equation) and Xⁿ¹ +Yⁿ²=Zⁿ³ (here simply referred as Beal´s equation). Starting from a bibliographical research on the Beal Conjecture, prime numbers and Fermat's Last Theorem, these equations were freely explored, searching for different aspects of their meanings. The developments on Beal Conjecture are divided into four parts: geometric illustrations; correspondence between the real solutions of Pythagoras´ equation and Fermat's equation; deduction of the transforms between the real solutions of Fermat's equation and the Beal´s equation; and analysis and discussion about the topic, including some examples. From the correspondent Pythagoras´ equation, if one of the terms A, B or C is taken as an integer reference basis, demonstrations enabled to show that the Beal Conjecture is correct if the remaining terms, when squared, are integers.