Definitive Proof of Beal's Conjecture

EasyChair Preprint no. 2163

Abstract

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.

Keyphrases: convenient numbers, Diophantine equation, divisibility, prime natural number, roots of polynomials of degree 3

@Booklet{EasyChair:2163,
  author = {Abdelmajid Ben Hadj Salem},
  title = {Definitive Proof of Beal's Conjecture},
  howpublished = {EasyChair Preprint no. 2163},

  year = {EasyChair, 2019}}