If Ax + By = Cz
If A, B, C, x, y, and z are all positive integers then A, B, and C should all have a common prime factor.
A common prime factor means that each of the numbers needs to be divisible by the same prime number. So 15, 10, and 5 all have a common prime factor of 5 (they're all divisible by the prime number 5).
So far, so simple, and it looks like something you would have solved in high school algebra.
But here's the problem. Mathematicians haven't ever been able to solve the Beale conjecture, with x, y, and z all being greater than 2.
For example, let's use our numbers with the common prime factor of 5 from before….
51 + 101 = 151
but
52 + 102 ≠ 152