solve the equation and win $1,000,000

[rudywoofs (Pam)]

(yes, this is a bona fide prize of $1,000,000)

So, can any of C/A's brilliant scientists/mathematicians solve this?

If Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor.

Background

Mathematicians have long been intrigued by Pierre Fermat's famous assertion that Ax + Bx = Cx is impossible (as stipulated) and the remark written in the margin of his book that he had a demonstration or "proof". This became known as Fermat's Last Theorem (FLT) despite the lack of a proof. Andrew Wiles proved the relationship in 1994, though everyone agrees that Fermat's proof could not possibly have been the proof discovered by Wiles. Number theorists remain divided when speculating over whether or not Fermat actually had a proof, or whether he was mistaken. This mystery remains unanswered though the prevailing wisdom is that Fermat was mistaken. This conclusion is based on the fact that thousands of mathematicians have cumulatively spent many millions of hours over the past 350 years searching unsuccessfully for such a proof.

It is easy to see that if Ax + Bx = Cx then either A, B, and C are co-prime or, if not co-prime that any common factor could be divided out of each term until the equation existed with co-prime bases. (Co-prime is synonymous with pairwise relatively prime and means that in a given set of numbers, no two of the numbers share a common factor.)

You could then restate FLT by saying that Ax + Bx = Cx is impossible with co-prime bases. (Yes, it is also impossible without co-prime bases, but non co-prime bases can only exist as a consequence of co-prime bases.)

Beyond Fermat's Last Theorem

No one suspected that Ax + By = Cz (note unique exponents) might also be impossible with co-prime bases until a remarkable discovery in 1993 by a Dallas, Texas number theory enthusiast by the name of D. Andrew “Andy” Beal. Andy Beal was working on FLT when he began to look at similar equations with independent exponents. He constructed several algorithms to generate solution sets but the very nature of the algorithms he was able to construct required a common factor in the bases. He began to suspect that co-prime bases might be impossible and set out to test his hypothesis by computer. Andy Beal and a colleague programmed 15 computers and after thousands of cumulative hours of operation had checked all variable values through 99. Many solutions were found: all had a common factor in the bases. While certainly not conclusive, Andy Beal now had sufficient reason to share his discovery with the world.

BEAL'S CONJECTURE: If Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor.

The Beal Conjecture

[cricket]

I don't think my minor in mathematics is going to help much. My brain hurts just thinking about that.

:)

[cricket]

RUDYWOOFS

so if x--y--z are greater than 2 then they are 3

A=1 B=1 C=2

1x3 + 1x3 = 2 x 3

6 = 6

this is my guess

so do I win smile smile smile

dgrimm60

+ =

[rudywoofs (Pam)]

I don't think that would be correct, Doug, since "Co-prime is synonymous with pairwise relatively prime and means that in a given set of numbers, no two of the numbers share a common factor."

But I'm not the one putting up the prize money, either! :)

[Bravus]

I think it's a copy-paste problem and it should read:

A^x + B^y = C^z