Diophantine equation with prime exponents

Question

Prove that the equation $2^p+3^p = a^n$ has no solutions, where $p$ is a prime number and $a,n > 1$ are integers.

The only thing that I proved is that $2^p+3^p$ is divisible by $5$ or that it equals $5$ modulo $p$ (Fermat's little theorem).

Answer 1

Hint.

Since you have proved $a$ must be divisible by $5$, reduce the equation modulo $25$ and see what you can deduce.

Answer 2

Since one is seeking $2^p + 3^p$ as a power, and $p$ is prime, then the only prime which yields a multiple of 25 for $2^p+3^p$ is $p=5$. But $275$ isn't a power of anything.

Any other prime yields a multiple of 5 and a non-multiple of 5.