Related to this question:
Can $f(n)=5^n+6^n+10^n$ with a non-negative integer $n$ be a perfect power ?
- It is not a perfect power for $0\le n\le 10^5$
- Analysis modulo $3$ reveals that a perfect square is impossible if $n\ge 2$ is even. Analysis modulo $8$ reveals that a perfect square is impossible , if $n\ge 1$ is odd. Hence the expression canoot be a perfect square.
What about cubes or higher perfect powers ? Considering the search limit, a perfect power of this form is extremely unlikely, but how can I make further progress towards a proof ?