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

If

`A`

^{x} + B^{y} = C^{z}

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)]`

,
and you get a new solution with ^{x}`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
`A`

^{x} + B^{y} = C^{z}

*Abstract:*

An intuitive look into the solutions to
`A`

(where ^{x} + B^{y} = C^{z}`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).