Journal of Integer Sequences, Vol. 13 (2010), Article 10.9.5

On the Fermat Periods of Natural Numbers

Tom Müller
Forschungsstelle für interdisziplinäre Geisteswissenschaft
Institut für philosophische Bildung
Alanus-Hochschule für Kunst und Gesellschaft
Villestr. 3
53347 Alfter bei Bonn
Kueser Akademie für europäische Geistesgeschichte
Gestade 18
54470 Bernkastel-Kues


Let b > 1 be a natural number and nN0. Then the numbers Fb,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 (Fb,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.

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(Concerned with sequences A102742 A128852.)

Received August 7 2010; revised version received October 11 2010; November 6 2010. Published in Journal of Integer Sequences, December 7 2010.

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