Archive
Special Issues Volume 2, Issue 5, October 2013, Page: 149-155
Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations
Leandro Torres Di Gregorio, Souza Marques EngineeringCollege, Rio de Janeiro, Brazil
Received: Aug. 20, 2013;       Published: Sep. 20, 2013
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 + γnn (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.
Keywords
Beal Conjecture, Fermat´s Last Theorem, Diophantine Equations, Number Theory, Prime Numbers
Leandro Torres Di Gregorio, Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations, Pure and Applied Mathematics Journal. Vol. 2, No. 5, 2013, pp. 149-155. doi: 10.11648/j.pamj.20130205.11
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