Müller, Tom On the Fermat periods of natural numbers. (English) Zbl 1208.11011 J. Integer Seq. 13, No. 9, Article 10.9.5, 12 p. (2010). 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. Reviewer: Jonas Šiaulys (Vilnius) Cited in 1 Document MSC: 11A41 Primes 11A51 Factorization; primality 11B83 Special sequences and polynomials 11N69 Distribution of integers in special residue classes 11Y05 Factorization Keywords:generalized Fermat number; Fermat period; elite prime number; anti-elite prime number Software:OEIS PDFBibTeX XMLCite \textit{T. Müller}, J. Integer Seq. 13, No. 9, Article 10.9.5, 12 p. (2010; Zbl 1208.11011) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: Elite primes: a prime number p is called elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic residues mod p. Anti-elite primes: a prime number p is called anti-elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic non-residues mod p.