## A Complete Proof of Beal Conjecture Followed by Numerical Examples

**Authors:** Abdelmajid Ben Hadj Salem

In 1997, Andrew Beal announced the following conjecture: \textit{Let $A, B,C, m,n$, and $l$ be positive integers with $m,n,l > 2$. If $A^m + B^n = C^l$ then $A, B,$ and $C$ have a common factor.} 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 of $A^m,B^n$ and $C^l$. We resolve $x^3-px+q=0$ and we obtain the three roots $x_1,x_2,x_3$ as functions of $p,q$ and a parameter $\theta$. Since $A^m,B^n,-C^l$ are the only roots of $x^3-px+q=0$, we discuss the conditions that $x_1,x_2,x_3$ are integers and have or not a common factor. Three numerical examples are given.

**Comments:** 46 Pages. In French, with minor corrections. Submitted to Journal Annales Scientifiques de l'Ecole Normale Supérieure. Comments welcome.

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### Submission history

[v1] 2016-12-17 11:51:55

[v2] 2017-04-10 05:59:50

[v3] 2017-06-18 11:27:33

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