The Beal Prize

Andrew Beal a self-made billionaire has announced the prize of one million dollars to the person who can prove the Beal conjecture.  The conjecture is similar to the famous last theorem of Pierre de Fermat.  Fermat’s last theorem, originally written in 1636 remained unsolved until 1995.    The theorem states that no three positive integers a, b, and c can satisfy the equation (aⁿ + bⁿ = cⁿ ), for any integer value of n>2.                  The Beal conjecture begins with the equation A ^X + B ^Y = C ^Z and states that if A, B, C, X, Y, and Z are all positive integers > 2 then A, B, and C must share a common factor.  The novice to mathematics might ask the question, what does all of this mean or more importantly why is this conjecture important?  This conjecture along with its predecessor Fermat’s last theorem is closely related to Diophantine equations.  Diophantine equations are a special set of equations named for Greek mathematician Diophantus of Alexandria.

The Diophantine equations all dealt with whole numbers and involved multiple unknown quantities. The most famous of these  equations is the Pythagorean Theorem, which is a² +b²= c ² where a and b are the two legs and c is the hypotenuse of a right triangle.  The Pythagorean equation shows the relationship of the lengths of all of the sides.  The real world applications of the Pythagorean Theorem are in engineering, architecture, cartography, and many other areas.  Diophantine equations all find their way into many separate real world applications.   The Beal conjecture is no exception the solution could lead to more applications, than just a simple intellectual curiosity.  Hopefully, this one doesn’t take 3 centuries to solve as did its predecessor.  

The proof of Fermat’s last Theorem was over 100 pages long and took 7 years of isolation for Sir Andrew John Wiles of Britain to complete. Will the Beal conjecture proof take as long? Now that the Fermat proof, which is closely related, is floating around, I propose that the solution to the Beal conjecture is within the Fermat proof or perhaps even an expansion or generalization of that proof.