Can we find all positive integers $a,b$ such that $a^{n}+b^{n}$ is an $(n+1)^{th}$ power? I think this question equivalent to solving the statement $$a^{n} + b^{n} = c^{n+1}$$ for $a,b,c \in \mathbb{N}$. But i don't know as to how i can solve this.
I attempted by subsituting $n$ as $n+1$ so that we get $$a^{n+1} + b^{n+1} = c^{n+2} = c \times c^{n+1}=(a^{n}+b^{n}) \cdot c$$
but it seems that this doesn't help!