The Beal Conjecture
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 Andrew Beal. 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. 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, Beal now had sufficient reason to share his discovery with the world.
Beal wrote many letters to mathematics periodicals and number theorists. Among the replies were two considered responses from number theorists. Dr. Harold Edwards from the department of mathematics at New York University and author of "Fermat's Last Theorem, a genetic introduction to algebraic number theory" confirmed that the discovery was unknown and called it "quite remarkable". Dr. Earl Taft from the department of mathematics at Rutgers University relayed Beal's discovery to an unidentified number theorist who was "an expert on Fermat's Last Theorem", according to Taft's response, and also confirmed that the discovery was unknown.
It is remarkable that in this day and age that such a simple and important part of number theory was unknown. Some mathematicians without a thorough knowledge of number theory incorrectly believe that the Beal discovery was evidenced in earlier work including that of Brun's papers of 1914-1915. This is clearly incorrect though Brun did examine some related forms and concepts in his work. The related ABC conjecture hypothesizes that only a finite number of solutions could exist. There is no known evidence of prior knowledge of Beal's conjecture and all references to it begin after Beal's 1993 discovery and subsequent dissemination of it.
By offering a cash prize for the proof or disproof of this important number theory relationship, Andrew Beal hopes to inspire young minds to think about the equation, think about winning the offered prize, and in the process become more interested in the wonderful study of mathematics. Information regarding the $50,000 cash prize that is held in trust by the American Mathematics Society can be obtained at the University of North Texas web site: http://www.math.unt.edu/~mauldin/beal.html.
Incidentally, Beal believes that the world has yet to adequately respond to Fermat's challenge to the English in 1657 regarding Pell's equation (see Edwards book referenced above - book section 1.9). Beal believes that Fermat may well have had a method of solution that was "not inferior to the more celebrated questions of geometry in respect of beauty, difficulty, or method of proof". Fermat claimed his method involved infinite descent and no known methods of solution use a descent. Furthermore, the continued fraction and cyclic methods of solution known today hardly qualify as beautiful or particularly difficult. For someone seeking to learn about or advance number theory, that's a great and fun place to start.
The Beal Conjecture is sometimes referred to as "Beal's conjecture", "Beal's problem" or the "Beal problem". Andrew Beal resides in Dallas, Texas where he owns and operates Beal Bank and Beal Aerospace Technologies, Inc.