NDLI: Fermat test with Gaussian base and Gaussian pseudoprimes

Content Provider	Springer Nature Link
Author	Oller Marcén, Antonio M. Grau, José María Rodríguez, Manuel Sadornil, Daniel
Copyright Year	2015
Abstract	The structure of the group (ℤ/nℤ)* 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 := {a + bi ∈ ℤ[i]/nℤ[i]: a 2 + b 2 ≡ 1 (mod n)}. 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.
Ending Page	982
Page Count	14
Starting Page	969
File Format	PDF
ISSN	00114642
e-ISSN	15729141
Journal	Czechoslovak Mathematical Journal
Issue Number	4
Volume Number	65
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-01-10
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Factorization; primality Ordinary Differential Equations Fermat test Counting solutions of Diophantine equations Analysis Convex and Discrete Geometry Arithmetic functions; related numbers; inversion formulas Mathematics Gaussian integer Mathematical Modeling and Industrial Mathematics pseudoprime
Content Type	Text
Resource Type	Article
Subject	Mathematics

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