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On elite primes of period four. (English) Zbl 1206.11007
A prime \(p\) is called elite (resp. anti-elite) if all sufficiently large Fermat numbers \(F_n = 2^{2^n}+1\) are quadratic non-residues (resp. residues) modulo \(p\). The sequence of \(F_n\) mod \(p\) is periodic, and the length \(L\) of its period is called the period of \(p\). Elite primes were introduced by A. Aigner [Monatsh. Math. 101, 85–93 (1986; Zbl 0584.10003)].
The present author proves that the numbers \(E_n = ((F_n-1)^5+1)/F_n\) (\(n\geq 2\)) have all their prime factors either elite or anti-elite with \(L=4\). He proves some theoretical results about these prime factors and has computed them up to \(n=7\); there are 6 elite and 10 anti-elite primes.

MSC:
11A15 Power residues, reciprocity
11A07 Congruences; primitive roots; residue systems
11A41 Primes
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[1] DOI: 10.1007/BF01298923 · Zbl 0584.10003 · doi:10.1007/BF01298923
[2] Bang A. S., Tidskrift Math. 5 pp 70–
[3] Chaumont A., J. Integer Seq. 9
[4] DOI: 10.1006/jnth.2002.2782 · Zbl 1026.11011 · doi:10.1006/jnth.2002.2782
[5] DOI: 10.1080/10586458.2006.10128955 · Zbl 1132.11005 · doi:10.1080/10586458.2006.10128955
[6] Müller T., J. Integer Seq. 10
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