Abstract
In this chapter we examine the problem of generalizing Lucas’ sequences from the perspective of linear recurrences. We first derive a number of features of such recurrences, culminating in a generalization of Lucas’ law of appearance. We follow this with an investigation of the properties of impulse sequences, particular cases of the more general linear recurrence. It turns out that the impulse sequences have many characteristics in common with the Lucas sequences. We next turn to the problem of generalizing the Lehmer sequencesLehmer sequences and how they may be used to produce necessary and sufficient tests of primality for numbers N of the form N = Apn + γ, where p is an odd prime, 2∣A, \(p\nmid A\), γ ∈{1, −1} and A < pn.
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Ballot, C.JC., Williams, H.C. (2023). Linear Recurrence Sequences and Further Generalizations. In: The Lucas Sequences. CMS/CAIMS Books in Mathematics, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-031-37238-4_7
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DOI: https://doi.org/10.1007/978-3-031-37238-4_7
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