Need help with a number theory proof

Is there a way to show that if there exists two natural numbers a and b coprime to each other, and a third prime p coprime to both a and b, that a^3 + b^3 cannot equal to some integer power of p with the exception of 1^3 + 1^3 = 2 and 1^3 + 2^3 = 3^2.

Re: Need help with a number theory proof

Alright, I got it

Assuming p,a,b are all relatively prime, and p is prime, then, it must be the case that an expression of the form .

If , then obviously , hence . This means that either or , but this contradicts the assumption that a,b,p are all mutually coprime. Hence, there cannot be any solutions to my equation under these constraints.