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A necessary and sufficient condition for the primality of Fermat numbers. (English) Zbl 0993.11002

Summary: We examine primitive roots modulo the Fermat numbers \(F_m=2^{2^m}+1\). We show that an odd integer \(n\geq 3\) is a Fermat prime if and only if the set of primitive roots modulo \(n\) is equal to the set of quadratic non-residues modulo \(n\). This result is extended to primitive roots modulo twice a Fermat number.

MSC:

11A41 Primes
11A07 Congruences; primitive roots; residue systems
11A51 Factorization; primality
11A15 Power residues, reciprocity
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