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?

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