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Keywords:
Fermat numbers; primitive roots; primality; Sophie Germain primes
Fermat numbers; primitive roots; primality; Sophie Germain primes
Summary:
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.
References:
[1] Burton, D. M.: Elementary Number Theory, fourth edition. McGraw-Hill, New York, 1998.
[2] Crandall, R. E., Mayer, E., Papadopoulos, J.: The twenty-fourth Fermat number is composite. Math. Comp. (submitted).
[3] Křížek, M., Chleboun, J.: A note on factorization of the Fermat numbers and their factors of the form $3h2^n+1$. Math. Bohem. 119 (1994), 437–445. MR 1316595
[4] Křížek, M., Luca, F., Somer, L.: 17 Lectures on Fermat Numbers. From Number Theory to Geometry. Springer, New York, 2001. MR 1866957
[5] Luca, F.: On the equation $\phi (|x^m-y^m|)=2^n$. Math. Bohem. 125 (2000), 465–479. MR 1802295 | Zbl 1014.11024
[6] Niven, I., Zuckerman, H. S., Montgomery, H. L.: An Introduction to the Theory of Numbers, fifth edition. John Wiley and Sons, New York, 1991. MR 1083765
[7] Szalay, L.: A discrete iteration in number theory. BDTF Tud. Közl. VIII. Természettudományok 3., Szombathely (1992), 71–91. (Hungarian) Zbl 0801.11011
[8] Wantzel, P. L.: Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas. J. Math. 2 (1837), 366–372.
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