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.

$\displaystyle \begin{align*} \gcd(a,b) &= 1 \\ \gcd(a,p) &= 1 \\ \gcd(b,p) &= 1 \\ a^3 + b^3 &\ne p^k \hspace{4mm} \mbox{for some } a,b,p,k \in \mathbb{N} \text{ and p is prime}\end{align*}$