The Beal Conjecture is a generalization of Fermat’s Last Theorem. D. Andrew Beal (Andy) was working on Fermat’s Theorem in 1993 when he began to look at similar equations, such as
Ax + By = Cz
with independent exponents x, y, and z. He constructed several algorithms to generate solution sets, but the very nature of the algorithms that he was able to construct required a common factor in the bases A, B, and C. He began to suspect that co-prime bases might be impossible and set out to test his hypothesis by computer. Extensive computational experiments were conducted in August 1993.
In the fall of 1994, Andy wrote letters about his work to approximately 50 mathematics journals and number theorists. Among the replies were two considered responses from respected number theorists confirming the novelty of the conjecture.
The first letter is from Professor Harold M. Edwards of the Department of Mathematics at New York University. Edwards authored Fermat's Last Theorem, a Genetic Introduction to Algebraic Number Theory, for which he was awarded the Leroy P. Steele Prize for Mathematical Exposition in 1980.
The second letter is from Professor Earl Taft of the Department of Mathematics at Rutgers University. He shared Andy Beal's discovery with Jerrold Tunnell, an expert on Fermat's Last Theorem. (Taft’s and Tunnell’s responses).
The Beal Conjecture and Prize were announced in an article that appeared in the December 1997 issue of Notices of the American Mathematical Society. A subsequent Letter to the Editor from R. Daniel Mauldin, author of the article, describes the early history of the Beal Conjecture.
The Beal Prize is described here.