Okay, so I have these equations:
$$ x = 1^p + 2^p + 3^p +... m^p $$ $$ x = 1^q + 2^q + 3^q +... n^q $$
How many possible values of positive integers $(m, n, p, q) $ are there, if any, such that the equations hold with $p,q,m, n> 1$ where $p \ne q$?
I tried approaching the problem by first considering the smaller values of $p$ and $q$ like $2,3$. and found no solution over the positive integers. I have no idea how to proceed with it. Any help will be appreciated.