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.
 &= 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*})
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.
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