Explore Beal's Conjecture: A Number Theory Challenge

[sh911098]

I’m a number theory lover but not an expert in the area. Recently, motivated by the report of Peter Norvig, Director of Research at Google, I’m interested in searching for counterexamples of Beal’s conjecture.

Billionaire banker Andrew Beal formulated this conjecture in 1993. For a proof or counterexample published in a refereed journal, Beal initially offered a prize of US dollar 5,000 in 1997, but raised it to 1,000,000 this Jun.

Beal’s conjecture is that if

A^x + B^y = C^z.

where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor.

The conjecture is very understandable. Below are a few positive examples
3³ + 6³ = 3⁵
where A = 3, B = 6, C = 3, x = 3, y = 3, z = 5; A, B, C have a common prime factor 3.
7⁶ + 7⁷ = 98³
where common prime factor is 7. Actually we always have
[A(A^m + B^m)]^m + [B(A^m + B^m)]^m = (A^m + B^m)^m+1
where A, B > 0, and m > 2.

However, followings are negative examples:
271³+2³×3⁵×73³ = 919³ = 776151559
3⁴×29³×89³+7³×11³×167³=2⁷×5⁴×359³=3518958160000
because the conjecture requires A, B, and C each has their own exponent, the two equations do not fulfill the requirement.Beal’s conjecture is a generation of Fermat’s last theorem, which has been proved by Andrew Willes in 1995. The theorem says

Aⁿ + Bⁿ = Cⁿ

has no solution of positive integers when n > 2. Every one understands the theorem is just a special case of Beal’s conjecture with x=y=z.

It took more than three hundreds years to get complete solution of Fermat’s last theorem.

How many years will be needed to have a proof or a counterexample of Beal’s conjecture? Nobody knows.

Do you have courage and interest to face the challenge? To me it is too hard to prove it. But I found it is not difficult to develop a program for the job, and I did one much fast than that used by Peter Norvig.

The purpose of this thread is to discuss basic idea and programming details of the project, and to associate as more as possible people, search as wide as possible.

 

[jvicens]

Thank you for sharing your interest in Beal's conjecture and your efforts to search for counterexamples. As a fellow number theory lover, I appreciate your enthusiasm and dedication to this topic.

Beal's conjecture is indeed a fascinating problem that has caught the attention of many mathematicians and scientists. It is often referred to as a "generalization" of Fermat's last theorem, as you mentioned in your post. However, it is worth noting that it is still an open problem and has not been proven or disproven yet.

As for how long it will take to find a proof or counterexample, it is difficult to say. It could take years, decades, or even longer. It ultimately depends on the progress and efforts of researchers and scientists like yourself. But regardless of the timeline, I believe that the pursuit of knowledge and understanding is always worth the effort.

I also appreciate your suggestion to use programming to aid in the search for counterexamples. In today's world, where technology is advancing at a rapid pace, it is important to utilize all available tools to tackle complex problems like this one. I encourage you to continue your work and to collaborate with others who share your interest in this topic.

I, too, am not an expert in this area, but I am always eager to learn and contribute in any way I can. I look forward to following this thread and participating in discussions about the basic ideas and programming details of this project. Let's continue to spread the word and involve more people in this exciting endeavor.