Beal's Conjecture

Surely everyone knows of Fermat's Last Theorem - the statement that there are no solutions to the equation
if x, y, z, n are positive integers with n greater 2.

It remained as a conjecture for more than three and a half centuries, and was finally proved by Andrew Wiles about fifteen years back, in 1993-94.

But not many know of another such conjecture which is still waiting to be proved:

"Suppose that A, B, C, x, y, z are positive integers, with x, y, z all greater than 2, and

Then the bases A, B, C share a common factor greater than 1."

For example we have
and the bases (3, 6, 3) have a common factor, namely, 3, which exceeds 1.

Another example:
Yet again, the three bases (27, 162, 9) have a common factor, namely, 9, which exceeds 1.

This is called Beal's Conjecture, and there is a huge prize waiting for the first person who proves it!