Given an odd prime number $p$. Find infinitely many triples $(m,k,n)$ of integers s.t. $k,m>1 $and $k$ and $m$ are not equal, fulfilling the following relations:
$$p|2^{n+\sqrt[k]{n}}-\sqrt[m]{n}$$ and $$p\nmid 2^{n+\sqrt[m]{n}}-\sqrt[k]{n}$$
I have found the following solutions to this problem:
For $p=3$: $(m,k,n)=(5,2,(6k-4)^{10})$ $\forall k\in \mathbb N$
For $p>3$: $(m,k,n)=(4,2,[(p-1)\cdot(pk+1)-1]^8)$ $\forall k\in \mathbb N$
To check that these triples indeed are solutions is easy to see by Fermat.
How to find all solutions to the above problem?