Let b > 1 be a natural number and n ∈ N₀. Then the numbers F_b,n := b²ⁿ + 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.