Fermat's Theorem states that aⁿ+bⁿ=cⁿ has no positive integer solutions for n>2. Since this puzzle has been already cracked, let's take the next level:

Prove that ⁿa+ⁿb=ⁿc has no integer solutions for n>1.

The "hyper power" ⁿx is defined recursively by ¹x:=x and ⁿ⁺¹x:=x^(nx), i.e. ²x=x^x, ³x=x^xx, and so on. Note that "hyper powers" are also called "double powers" and may be written as x^^n.