On some new results for the generalised Lucas sequences

Dorin Andrica; Ovidiu Bagdasar; George Cătălin Ţurcaş

doi:10.2478/auom-2021-0002

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On some new results for the generalised Lucas sequences

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 29 (2021): Issue 1 (March 2021)

By: Dorin Andrica, Ovidiu Bagdasar and George Cătălin Ţurcaş

Open Access

|Apr 2021

Abstract

In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

References

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Articles in this issue

DOI: https://doi.org/10.2478/auom-2021-0002 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

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Language: English

Page range: 17 - 36

Submitted on: Apr 1, 2020

Accepted on: May 25, 2020

Published on: Apr 13, 2021

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Generalised Lucas sequence,

Generalised Pell-Lucas sequence,

Pell equation,

Special Pell Equation,

Negative Pell equation

Related subjects:

Mathematics,

General mathematics

© 2021 Dorin Andrica, Ovidiu Bagdasar, George Cătălin Ţurcaş, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 29 (2021): Issue 1 (March 2021)