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On the Fermat Periods of Natural Numbers
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Tom Müller

Forschungsstelle für interdisziplinäre Geisteswissenschaft

Institut für philosophische Bildung

Alanus-Hochschule für Kunst und Gesellschaft

Villestr. 3

53347 Alfter bei Bonn

Germany

and

Kueser Akademie für europäische Geistesgeschichte

Gestade 18

54470 Bernkastel-Kues

Germany

**Abstract:**

Let *b* > 1 be a natural number
and *n* ∈ **N**_{0}. Then the numbers
*F*_{b,n} := *b*^{2n} + 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.

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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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