If you can prove the conjecture to be true (or false), you will win
a million dollar prize.
One way to prove it false is to come up with a counterexample.
A valid counterexample must have these properties:

A^{x} + B^{y} must equal C^{z}

A, B, C, x, y, z must all be positive integers

x, y, z must all be greater than 2

A, B, C must have no common factor (other than 1).

Check your counterexample here:

Enter values for A, x, B, y, C, z and hit the Check button.