A necessary and sufficient condition for the primality of Fermat numbers

Michal Křížek; Lawrence Somer

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 3, page 541-549
  • ISSN: 0862-7959

Abstract

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We examine primitive roots modulo the Fermat number F m = 2 2 m + 1 . We show that an odd integer n ≥ 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.

How to cite

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Křížek, Michal, and Somer, Lawrence. "A necessary and sufficient condition for the primality of Fermat numbers." Mathematica Bohemica 126.3 (2001): 541-549. <http://eudml.org/doc/248886>.

@article{Křížek2001,
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.},
author = {Křížek, Michal, Somer, Lawrence},
journal = {Mathematica Bohemica},
keywords = {Fermat numbers; primitive roots; primality; Sophie Germain primes; Fermat numbers; primitive roots; primality; Sophie Germain primes},
language = {eng},
number = {3},
pages = {541-549},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A necessary and sufficient condition for the primality of Fermat numbers},
url = {http://eudml.org/doc/248886},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Křížek, Michal
AU - Somer, Lawrence
TI - A necessary and sufficient condition for the primality of Fermat numbers
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 3
SP - 541
EP - 549
AB - 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.
LA - eng
KW - Fermat numbers; primitive roots; primality; Sophie Germain primes; Fermat numbers; primitive roots; primality; Sophie Germain primes
UR - http://eudml.org/doc/248886
ER -

References

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  1. Elementary Number Theory, fourth edition, McGraw-Hill, New York, 1998. (1998) 
  2. The twenty-fourth Fermat number is composite, Math. Comp. (submitted). 
  3. A note on factorization of the Fermat numbers and their factors of the form 3 h 2 n + 1 , Math. Bohem. 119 (1994), 437–445. (1994) MR1316595
  4. 17 Lectures on Fermat Numbers. From Number Theory to Geometry, Springer, New York, 2001. (2001) MR1866957
  5. On the equation φ ( | x m - y m | ) = 2 n , Math. Bohem. 125 (2000), 465–479. (2000) Zbl1014.11024MR1802295
  6. An Introduction to the Theory of Numbers, fifth edition, John Wiley and Sons, New York, 1991. (1991) MR1083765
  7. A discrete iteration in number theory, BDTF Tud. Közl. VIII. Természettudományok 3., Szombathely (1992), 71–91. (Hungarian) (1992) Zbl0801.11011
  8. 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. (1837) 

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