On some new results for the generalised Lucas sequences

Andrica, Dorin; Bagdasar, Ovidiu; Ţurcaş, George Cătălin

doi:10.2478/auom-2021-0002

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

[1] T. Andreescu, D. Andrica, Equations with Solution in terms of Fibonacci and Lucas Sequences, An. Şt. Univ. Ovidius Constanţa 22(3) (2014), 5–12.10.2478/auom-2014-0046
Search in Google Scholar Back to article
[2] T. Andreescu, D. Andrica, Quadratic Diophantine Equations, Springer, New York (2015).10.1007/978-0-387-54109-9
Search in Google Scholar Back to article
[3] D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems, Springer (2020)10.1007/978-3-030-51502-7
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[6] Y. Bugeaud, M. Mignotte, Y. Roy, On the Diophantine equationxn-1x-1=yq{{{x^{n - 1}}} \over {x - 1}} = {y^q}, Pac. J. Math. 193(2), 257–268 (2000)10.2140/pjm.2000.193.257
Search in Google Scholar Back to article
[7] Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. Math. 163, 969 – 1018 (2006)
Search in Google Scholar Back to article
[8] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer, New York (1998)
Search in Google Scholar Back to article
[9] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, Inc., Hoboken, NJ, USA (2001)10.1002/9781118033067
Search in Google Scholar Back to article
[10] D. H. Lehmer, An extended theory of Lucas functions, Ann. Math., 2nd Ser. 2(3), 419–448 (1930)10.2307/1968235
Search in Google Scholar Back to article
[11] W. Ljunggren, Noen setninger om ubestemte likninger av formen (xn −1)/(x − 1) = yq, Norsk. Mat. Tidsskr. 25, 17–20 (1943)
Search in Google Scholar Back to article
[12] L. Mordell, Diophantine equations, Volume 30, Academic Press (1969).
Search in Google Scholar Back to article
[13] The On-Line Encyclopedia of Integer Sequences, http://oeis.org, OEIS Foundation Inc. 2011.
Search in Google Scholar Back to article
[14] S. Schuster, M. Fitchner, S. Sasso, Use of Fibonacci numbers in lipidomics - Enumerating various classes of fatty acids, Nature Sci. Rep. 7, 39821, doi:10.1038/srep39821 (2017).10.1038/srep39821522315828071669
Search in Google Scholar Back to article

On some new results for the generalised Lucas sequences

Abstract

Paradigm

My account