Fermat test with Gaussian base and Gaussian pseudoprimes

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

doi:10.1007/s10587-015-0221-2

José María Grau, Manuel Rodríguez + Show 2 more

Open Access

https://doi.org/10.1007/s10587-015-0221-2

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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 Gn:= {a + bi ∈ ℤ[i]/nℤ[i]: a2 + b2 ≡ 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.

Highlights

Most of the classical primality tests are based on Fermat’s little theorem: let p be a prime number and let a be an integer such that p ∤ a, ap−1 ≡ 1
The problem is that the converse is false and there exists composite numbers n such that an−1 ≡ 1 for some a coprime to n
We present the concepts of Gaussian pseudoprime and Gaussian Carmichael numbers presenting an explicit Korselt’s criterion

Summary

Introduction

Most of the classical primality tests are based on Fermat’s little theorem: let p be a prime number and let a be an integer such that p ∤ a, ap−1 ≡ 1 (mod p). The problem is that the converse is false and there exists composite numbers n such that an−1 ≡ 1 (mod n) for some a coprime to n In this situation n is called pseudoprime with respect to base a Associated with the subgroup (Z/nZ)⋆ we can define the well-know Euler’s totient function and Carmichael’s lambda function which are defined in the following way: φ(n) := |(Z/nZ)⋆|, λ(n) := exp(Z/nZ)⋆ It seems reasonable (and natural) to extend these ideas to other general groups Gn. It seems reasonable (and natural) to extend these ideas to other general groups Gn This extension leads to composite/primality tests according to the following steps: 1◦) Compute f (n) = |Gn| under the assumption that n is prime. This strength is possible due to a relationship with 1-Williams numbers [21] that we make explicit

Preliminaries

Gaussian Fermat pseudoprimes

Gaussian Carmichael and cyclic numbers

G-Lehmer’s totient problem and G-Giuga’s conjecture