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.
Download: PDF
[v1] 2016-12-17 11:51:55
[v2] 2017-04-10 05:59:50
[v3] 2017-06-18 11:27:33
Unique-IP document downloads: 181 times
Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.