| Title:
|
A necessary and sufficient condition for the primality of Fermat numbers (English) |
| Author:
|
Křížek, Michal |
| Author:
|
Somer, Lawrence |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
126 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
541-549 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Fermat numbers |
| Keyword:
|
primitive roots |
| Keyword:
|
primality |
| Keyword:
|
Sophie Germain primes |
| MSC:
|
11A07 |
| MSC:
|
11A15 |
| MSC:
|
11A41 |
| MSC:
|
11A51 |
| idZBL:
|
Zbl 0993.11002 |
| idMR:
|
MR1970256 |
| DOI:
|
10.21136/MB.2001.134197 |
| . |
| Date available:
|
2009-09-24T21:53:50Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134197 |
| . |
| Reference:
|
[1] Burton, D. M.: Elementary Number Theory, fourth edition.McGraw-Hill, New York, 1998. |
| Reference:
|
[2] Crandall, R. E., Mayer, E., Papadopoulos, J.: The twenty-fourth Fermat number is composite.Math. Comp. (submitted). |
| Reference:
|
[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 |
| Reference:
|
[4] Křížek, M., Luca, F., Somer, L.: 17 Lectures on Fermat Numbers. From Number Theory to Geometry.Springer, New York, 2001. MR 1866957 |
| Reference:
|
[5] Luca, F.: On the equation $\phi (|x^m-y^m|)=2^n$.Math. Bohem. 125 (2000), 465–479. Zbl 1014.11024, MR 1802295 |
| Reference:
|
[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 |
| Reference:
|
[7] Szalay, L.: A discrete iteration in number theory.BDTF Tud. Közl. VIII. Természettudományok 3., Szombathely (1992), 71–91. (Hungarian) Zbl 0801.11011 |
| Reference:
|
[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. |
| . |
