×

Repdigits in generalized Pell sequences. (English) Zbl 07285963

Summary: For an integer \(k\ge 2\), let \((P_n^{(k)})_n\) be the \(k\)-generalized Pell sequence which starts with \(0,\ldots ,0,1\) (\(k\) terms) and each term afterwards is given by the linear recurrence \(P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}\). In this paper, we find all \(k\)-generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence \((P_n^{(2)})_n\).

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11J86 Linear forms in logarithms; Baker’s method
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Baker, A.; Davenport, H., The equations \(3x^2-2=y^2\) and \(8x^2-7=z^2\), Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137
[2] Bravo, J. J.; Gómez, C. A.; Luca, F., Powers of two as sums of two \(k-\) Fibonacci numbers, Miskolc Math. Notes 17 (1) (2016), 85-100
[3] Bravo, J. J.; Herrera, J. L.; Luca, F., On a generalization of the Pell sequence, doi:10.21136/MB.2020.0098-19 on line in Math. Bohem
[4] Bravo, J. J.; Luca, F., On a conjecture about repdigits in \(k-\) generalized Fibonacci sequences, Publ. Math. Debrecen 82 (3-4) (2013), 623-639
[5] Bravo, J. J.; Luca, F., Repdigits in \(k\)-Lucas sequences, Proc. Indian Acad. Sci. Math. Sci. 124 (2) (2014), 141-154
[6] Dujella, A.; Pethö, A., A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (195) (1998), 291-306
[7] Faye, B.; Luca, F., Pell and Pell-Lucas numbers with only one distinct digits, Ann. of Math. 45 (2015), 55-60
[8] Kiliç, E., The Binet formula, sums and representations of generalized Fibonacci \(p\)-numbers, European J. Combin. 29 (2008), 701-711
[9] Kiliç, E., On the usual Fibonacci and generalized order \(-k\) Pell numbers, Ars Combin 109 (2013), 391-403
[10] Kiliç, E.; Taşci, D., The generalized Binet formula, representation and sums of the generalized order \(-k\) Pell numbers, Taiwanese J. Math. 10 (6) (2006), 1661-1670
[11] Koshy, T., Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics, Wiley-Interscience Publications, New York, 2001
[12] Luca, F., Fibonacci and Lucas numbers with only one distinct digit, Port. Math. 57 (2) (2000), 243-254
[13] Marques, D., On \(k\)-generalized Fibonacci numbers with only one distinct digit, Util. Math. 98 (2015), 23-31
[14] Matveev, E. M., An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (6) (2000), 125-180, translation in Izv. Math. 64 (2000), no. 6, 1217-1269
[15] Normenyo, B.; Luca, F.; Togbé, A., Repdigits as sums of three Pell numbers, Period. Math. Hungarica 77 (2) (2018), 318-328
[16] Normenyo, B.; Luca, F.; Togbé, A., Repdigits as sums of four Pell numbers, Bol. Soc. Mat. Mex. (3) 25 (2) (2019), 249-266
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.