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Journal of Integer Sequences, Vol. 13 (2010), Article 10.9.5 On the Fermat Periods of Natural Numbers

By Tom Müller and Forschungsstelle Für Interdisziplinäre Geisteswissenschaft


Let b> 1 be a natural number and n ∈ N0. 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.

Year: 2011
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