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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
##### Keywords:
Fermat numbers; quadratic residues; elite primes
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##### References:
 [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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