Beal's Conjecture

Beal's Conjecture


Andrew Beal has conjectured the following neat generalization of Fermat's Last Theorem:

If
       Ax + By = Cz

where A,B,C,x,y,z are positive integers and x,y,z > 2, then A, B, C must all share a common factor greater than 1, i.e. gcd(A,B,C) > 1.

No numerical counter-examples have been found yet. Is it true? By embedding the problem in a larger space will it lead to a simpler proof of Fermat's Last Theorem?

It is easy to show that if Beal's Conjecture is true then it implies Fermat's Last Theorem: since for x = y = z > 2 , assume a solution A, B, C and then divide the equation the common factor= [gcd(A,B,C)]x, and you get a new solution with gcd(A',B',C') = 1, in contradiction of Beal's conjecture. Thus there can be no solutions for x = y = z > 2. This was conjectured by Fermat in the 1600's and proved recently by Wiles. Is the rest of Beal's conjecture true? i.e. with x y z not all equal?

More information may be found at Prof. Daniel Mauldin's website.


Here is a draft of a fun calculation I did roughly related to the above conjecture. It basically demonstrates the extreme danger involved in permitting a computational physicist to dabble at number theory!

A Probabilistic Model of the Solutions to Ax + By = Cz

Abstract:
An intuitive look into the solutions to Ax + By = Cz (where A, B, C, x, y, and z are positive integers) is done by creating a heuristic model of the "probability" of solving the equation. The resulting probability distributions are intended to guide computational searches for counter-examples to Beal's conjecture, which is: A, B, C always have a common factor if x, y, z > 2. This simple treatment also actually reproduces some known results about the finiteness of the number of solutions for small x, y, and z.


Here are some notes on a similar calculation I made for Goldbach's conjecture (every even number greater than 4 can be expressed as the sum of two odd primes).