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Fermat test with Gaussian base and Gaussian pseudoprimes

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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 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.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Oviedo, Avenida Calco Sotelo s/n, 33007, Gijón, Spain

    José María Grau

  2. Centro Universitario de la Defensa de Zaragoza, Ctra. de Huesca s/n, 50090, Zaragoza, Spain

    Antonio M. Oller-Marcén

  3. Avenida Ramón Ferreriro 19, 27002, Lugo, Spain

    Manuel Rodríguez

  4. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida de los Castros s/n, 39005, Santander, Spain

    Daniel Sadornil

Authors
  1. José María Grau
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  2. Antonio M. Oller-Marcén
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  3. Manuel Rodríguez
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  4. Daniel Sadornil
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Corresponding author

Correspondence to José María Grau.

Additional information

D. Sadornil is partially supported by the Spanish Government under projects MTM2010-21580-C02-02 and MTM2010-16051.

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Grau, J.M., Oller-Marcén, A.M., Rodríguez, M. et al. Fermat test with Gaussian base and Gaussian pseudoprimes. Czech Math J 65, 969–982 (2015). https://doi.org/10.1007/s10587-015-0221-2

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

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