The Last Primetime
\[\large x^n + y^n = z^n\]
All variables in this equation represent positive integers. Moreover, \(n > 1\).
For what integral value(s) of \(n\) does the equation have solutions?
Submit answer as sum of possible values of \(n\), if finite solutions.
Submit \(666\) , if no solution possible.
submit \(999\) , if there are infinite solutions.