Weak Pseudoprimality Associated with the Generalized Lucas Sequences

Dorin Andrica (), Ovidiu Bagdasar () and Michael Th. Rassias ()
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Dorin Andrica: Babeş-Bolyai University
Ovidiu Bagdasar: University of Derby
Michael Th. Rassias: University of Zurich

A chapter in Approximation and Computation in Science and Engineering, 2022, pp 53-75 from Springer

Abstract: Abstract Pseudoprimes are composite integers which share properties of the prime numbers, and they have applications in many areas, as, for example, in public-key cryptography. Numerous types of pseudoprimes are known to exist, many of them defined by linear recurrent sequences. In this material, we present some novel classes of pseudoprimes related to the generalized Lucas sequences. First, we present some arithmetic properties of the generalized Lucas and Pell–Lucas sequences and review some classical pseudoprimality notions defined for Fibonacci, Lucas, Pell, and Pell–Lucas sequences and their generalizations. Then we define new notions of pseudoprimality which do not involve the use of the Jacobi symbol and include many classical pseudoprimes. For these, we present associated integer sequences recently added to the Online Encyclopedia of Integer Sequences, identify some key properties, and propose a few conjectures.

Keywords: Pseudoprime; Generalized Lucas sequence; Generalized Pell–Lucas sequence; Generalized Bruckman–Lucas pseudoprime; Weak generalized Lucas pseudoprime; Weak generalized Bruckman–Lucas pseudoprime; Primary 11A51; Secondary 11B39; 11Y50 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/978-3-030-84122-5_4

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