@JyrkiLahtonen I am not sure to which extent this might be of interest to you. You posted an answer to this (now deleted) question:
How we can deal with this equation $a^n+b^n=c^n$ if it was given to have solutions in primes numbers not integers numbers?. If you feel like that, you might repost that answer to my question:
No solutions of $x^n+y^n=z^n$ such that $x$, $y$, $z$ are primes.