Geodesic
On the Fermat periods of natural numbers
Journal of integer sequences, Tome 13 (2010) no. 9.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 11A41, 11A51, 11N69, 11Y05
Keywords: generalized Fermat number, elite prime number, anti-elite prime number, Fermat period 
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     author = {M\"uller, Tom},
     title = {On the {Fermat} periods of natural numbers},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {13},
     number = {9},
     year = {2010},
     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/