BEAL’S CONJECTURE

A^x+B^y = C^z => hcf (A,B,C) > 1

Proof:

Basically a corollary of the real Fermat’s Last Theorem (see attached 3 pages), with the following slight variation(s) in reasoning:

  1. Throughout, consider the equation:

    2. 1.A^x + B^(y-x).B^x = C^(z-x).C^x

    as opposed to the original Fermat equation:

    A^x + B^x = C^x

  2. Apply the same reasoning (see attached 3 pages) to deduce that there is no smallest C
  3. But hcf (A,B,C) = 1 => there is a smallest (A, B &) C
  4. (2) and (3) together => Beal’s Conjecture.