Weak Pseudoprimality Associated with the Generalized Lucas Sequences

Abstract

AbstractPseudoprimes 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.KeywordsPseudoprimeGeneralized Lucas sequenceGeneralized Pell–Lucas sequenceGeneralized Bruckman–Lucas pseudoprimeWeak generalized Lucas pseudoprimeWeak generalized Bruckman–Lucas pseudoprime2010 AMS Subject ClassificationPrimary 11A51Secondary 11B3911Y50