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Primes modulo which almost all Fermat numbers are primitive roots. (English) Zbl 1234.11006
Let $$p$$ be an odd prime number. The sequence of Fermat numbers $$F_n=2^{2^n}+1$$ is periodic modulo $$p$$ with a preperiod of length $$s$$, and a period of length $$L$$. Both numbers $$s$$ and $$L$$ can be easily computed in terms of the order of $$2$$ modulo the prime $$p$$. An elite prime is a prime number $$p$$ such that all $$L$$ residues appearing in the period are quadratic nonresidues. The author calls an elite prime to be ultra-elite if additionally either all these residues are primitive roots modulo $$p$$, or all these residues are non-primitive roots modulo $$p$$. The author studies necessary and sufficient conditions for a prime $$p$$ to be ultra-elite in the case when $$L=4$$, condition which is equivalent to the fact that the order of $$2$$ modulo $$p$$ is of the form $$2^s\times 5$$. The main questions addressed are under which conditions the four residues in the period are not distinct, or what else can be said if one of them equals to $$1$$. For example, it is shown that the number of primitive roots among these residues can never be $$1$$ or $$2$$. The proofs involve elementary computations with the fifth primitive root of unity modulo $$p$$. Some numerical examples of ultra-elite primes are presented and some open questions are formulated.
##### MSC:
 11A15 Power residues, reciprocity 11A41 Primes
##### Keywords:
Fermat numbers; elite primes
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