I have the polynomial $f(x) = x^2-x+1$ and I am wondering if there is a positive prime value $p$ such that $f(p),f^2(p),f^3(p)\dots$ are all prime.
I have ran some computer simulations and I feel like the answer should be "no" ( because looking at the map $x^2-x+1 \bmod p$ I get that the expected number of prime divisors of the first $M$ values should be larger than $M$). I feel that my analysis is not very good however.
Does anyone know an approach which could be more fruitful?
Note: $f^2(x) = f(f(x))$
Edit: I would be happy with any sort of reference regarding a polynomial staying inside the primes under iteration, if there isn't anything particularly useful which can be said about this sort of thing I understand that as well).