Fermat Prime?

Find all triples \((p,n,k)\) of positive integers, where \(p\) is a Fermat's Prime, satisfying

\[\large{p^n + n = (n+1)^k}\]


Observation: a Fermat's Prime is a prime number of the form \(2^{\alpha} + 1\), for \(\alpha\) positive integer.

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