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That would depend on $k$. I'm assuming you want all variables positive integers with $m>1$ and $n>1$. It may have been proved that for any given $k$ there are only finitely many solutions.

Edit: in view of the other answers, it appears my "may have been proved" was overly optimistic.

show/hide this revision's text 1

That would depend on $k$. I'm assuming you want all variables positive integers with $m>1$ and $n>1$. It may have been proved that for any given $k$ there are only finitely many solutions.