Summary: Let $b > 1$ be a natural number and $n \in N_{0}$. Then the numbers $F_{b,n} := b^{2^{n}} + 1$ form the sequence of generalized Fermat numbers in base $b$. It is well-known that for any natural number $N$, the congruential sequence $(F_{b,n} (mod N))$ is ultimately periodic. We give criteria to determine the length of this Fermat period and show that for any natural number $L$ and any $b > 1$ the number of primes having a period length $L$ to base $b$ is infinite. From this we derive an approach to find large non-Proth elite and anti-elite primes, as well as a theorem linking the shape of the prime factors of a given composite number to the length of the latter number's Fermat period.

11A41, 11A51, 11N69, 11Y05

generalized Fermat number, elite prime number, anti-elite prime number, Fermat period