Number Theory

    

Proof of Fermat's Last Theorem, Beal Conjecture, and Catalan's Conjecture [?]

Authors: Mi Zhou

This paper investigates the non-existence of positive integer solutions for equationsrelated to Fermat's Last Theorem, Beal Conjecture, and Catalan's Conjecture. For (4n1 + 1)n+(4n2 + 1)n=(4n3)n , expanding the left-hand side yields a term of the form 4nu2032+2, while the right-hand side is 4nu2032u2032, demonstrating the equation's invalidity. Fermat's Last Theorem (xn + y n=z n with n > 2) was proven by Wiles using highly complex methods. The generalized Fermat equation (x p + y q=z r) extends this, with Beal Conjecture positing no positive integer solutions when x, y, and z arecoprime—a problem yet unresolved. Catalan's Conjecture (A m=B n + 1) asserts no solutions exist beyond 3 2=2 3 + 1, proven by Preda Mihăilescu through intricate means. This study employs concise modular arithmetic to address all three conjectures.

Comments: 5 Pages.

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Submission history

[v1] 2025-08-05 20:38:56

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