Number Theory

    

A Complete Proof of Beal's Conjecture Final Version

Authors: Abdelmajid Ben Hadj Salem

In 1997, Andrew Beal announced the following conjecture: Let A, B, C, m, n, and l bepositive integers with m, n, l > 2. If A^m + B^n = C^l then A, B, and C have a commonfactor. We begin to construct the polynomial P(x) = (x − A^m)(x − B^n)(x + C^l) =x^3 − px + q with p, q integers depending on A^m, B^n and C^l. We resolve x^3 − px + q = 0and we obtain the three roots x_1, x_2, x_3 as functions of p and a parameter θ. SinceA^m, B^n, −C^l are the only roots of x^3 − px + q = 0, we discuss the conditions thatx_1, x_2, x_3 are integers and have or do have not a common factor. Three numericalexamples are given.

Comments: 58 Pages. It is the final version of my proof of Beal's conjecture. Submitted to the journal Mathematika. Comments welcome.

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Submission history

[v1] 2019-10-20 11:58:01
[v2] 2023-08-21 13:13:18
[v3] 2024-11-29 10:26:41

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