Geodesic
On the Fermat periods of natural numbers
Journal of integer sequences, Tome 13 (2010) no. 9
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.
Classification : 11A41, 11A51, 11N69, 11Y05
Keywords: generalized Fermat number, elite prime number, anti-elite prime number, Fermat period 
@article{JIS_2010__13_9_a4,
     author = {M\"uller,  Tom},
     title = {On the {Fermat} periods of natural numbers},
     journal = {Journal of integer sequences},
     year = {2010},
     volume = {13},
     number = {9},
     zbl = {1208.11011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_9_a4/}
}
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Müller,  Tom. On the Fermat periods of natural numbers. Journal of integer sequences, Tome 13 (2010) no. 9. http://geodesic.mathdoc.fr/item/JIS_2010__13_9_a4/