On some new results for the generalised Lucas sequences-Reference-Cited by-同舟云学术

On some new results for the generalised Lucas sequences

Published:2021-03-01 Issue:1 Volume:29 Page:17-36
ISSN:1844-0835
Container-title:Analele Universitatii "Ovidius" Constanta - Seria Matematica
language:en
Short-container-title:

Author:

Andrica Dorin¹,Bagdasar Ovidiu²,Ţurcaş George Cătălin¹

Affiliation:

1. Faculty of Mathematics and Computer Science , Babeş-Bolyai University of Cluj-Napoca , Kogălniceanu Street, Nr. 1, 400084 Cluj-Napoca , Romania .

2. Department of Electronics, Computing and Mathematics , University of Derby , Kedleston Road, Derby DE22 1GB, England, UK .

Abstract

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.

Publisher

Walter de Gruyter GmbH

Link

https://www.sciendo.com/pdf/10.2478/auom-2021-0002

Reference14 articles.

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2. [2] T. Andreescu, D. Andrica, Quadratic Diophantine Equations, Springer, New York (2015).10.1007/978-0-387-54109-9

3. [3] D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems, Springer (2020)10.1007/978-3-030-51502-7

4. [4] D. Andrica, O. Bagdasar, On some arithmetic properties of the generalised Lucas sequences, Med. J. Math. 18, Article 47 (2021)10.1007/s00009-020-01653-w

5. [5] D. Andrica, V. Crişan, F. Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab J. Math. Sci. 24(1), 9–15 (2018)10.1016/j.ajmsc.2017.06.002

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