MATHEMATICA BOHEMICA, Vol. 126, No. 3, pp. 541-549 (2001)

A necessary and sufficient condition for the primality of Fermat numbers

Michal Krizek, Lawrence Somer

Michal Krizek, Mathematical Institute, Academy of Sciences, Zitna 25, CZ-115 67 Praha 1, Czech Republic, e-mail:; Lawrence Somer, Department of Mathematics, Catholic University of America, Washington, D.C. 20064, U.S.A., e-mail:

Abstract: We examine primitive roots modulo the Fermat number $F_m=2^{2^m}+1$. We show that an odd integer $n\ge 3$ 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.

Keywords: Fermat numbers, primitive roots, primality, Sophie Germain primes

Classification (MSC2000): 11A07, 11A15, 11A51

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