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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.
11A15 Power residues, reciprocity
11A41 Primes
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