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On the Fermat periods of natural numbers. (English) Zbl 1208.11011

Let \(b>1\) be a natural number and let \(n\in \{0,1,2,\dots\}\). The numbers \(F_{b,n}=b^{2^{n}}+1\) form the sequence of generalized Fermat numbers in base \(b\). The numbers \(F_{b,n}\) fulfill some recurrence relation. So the congruential sequence \(\{F_{b,n}\pmod N\}\) is ultimately periodic (with a period length \(L_b(N)\)) for any fixed natural number \(N\). Author gives criteria to determine the length of this Fermat period \(L_b(N)\) and shows that for any natural number \(L\) and any natural \(b>1\) the number of primes having a period length \(L\) in base \(b\) is infinite. From this assertion author derives 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 of Fermat period.

MSC:

11A41 Primes
11A51 Factorization; primality
11B83 Special sequences and polynomials
11N69 Distribution of integers in special residue classes
11Y05 Factorization

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