Fermat test with Gaussian base and Gaussian pseudoprimes

Title:

Fermat test with Gaussian base and Gaussian pseudoprimes

Creator:

Grau, José María, Oller-Marcén, Antonio M, Rodríguez, Manuel, and Sadornil, Daniel

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:dd73498e-4739-4b16-b742-9460053fd6c3
uuid:dd73498e-4739-4b16-b742-9460053fd6c3
issn:0011-4642

Subject:

Gaussian integer, Fermat test, pseudoprime, 13, and 51

Type:

model:article and TEXT

Format:

print, bez média, and svazek

Description:

The structure of the group (\mathbb{Z}/n\mathbb{Z})* and Fermat’s little theorem are the basis for some of the best-known primality testing algorithms. Many related concepts arise: Euler’s totient function and Carmichael’s lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer’s totient problem, Giuga’s conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group G_{n}:=\left \{ a+bi\in \mathbb{Z}\left [ i\right ]:a^{2}+b^{2}\equiv 1\left ( mod n \right ) \right \}. In particular, we characterize Gaussian Carmichael numbers via a Korselt’s criterion and present their relation with Gaussian cyclic numbers. Finally, we present the relation between Gaussian Carmichael number and 1-Williams numbers for numbers n ≡ 3 (mod 4). There are also no known composite numbers less than 1018 in this family that are both pseudoprime to base 1 + 2i and 2-pseudoprime., José María Grau, Antonio M. Oller-Marcén, Manuel Rodríguez, Daniel Sadornil., and Obsahuje seznam literatury

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

969-982

Source:

Czechoslovak Mathematical Journal | 2015 Volume:65 | Number:4

Harvested from:

CDK

Metadata only:

false