I believe the conjecture, called the Beal Conjecture, after the banker and
amateur mathematician that discovered it, is as follows.
Given A,B,C,x,y,z whole numbers, x,y,z > 2. If A^x + B^y = C^z, then A, B, C
have a common factor.
The only examples I could generate in a few minutes of playing with a
spreadsheet are 32^3 + 8^5 = 16^4, and 15^4 + 15^5 = 30^4.
If you let the exponents include 2, then 13^2 + 3^3 = 14^2 is a
counterexample. Proving the Beal Conjecture would be another way to prove
Fermat's Last Theorem.
Anyway you may want to check out the article yourself.
Phil Mahler
Middlesex CC
Bedford, MA
p.s. As I was searching for examples, with a TV on nearby, Star Trek - Next
Generation came on - an old episode obviously. At the beginning, the captain
was playing with Fermat's Last Theorem on his computer.
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