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Keywords:
generalized Pell numbers; repdigits; linear forms in logarithms; reduction method
generalized Pell numbers; repdigits; linear forms in logarithms; reduction method
Summary:
For an integer $k\ge 2$, let $({n})_n$ 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 ${n} = 2{n-1}+{n-2}+\cdots +{n-k}$. 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 $(P_n^{(2)})_n$.
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. MR 0248079
[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. DOI 10.18514/MMN.2016.1505 | MR 3527869
[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. DOI 10.21136/MB.2020.0098-19
[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. DOI 10.5486/PMD.2013.5390 | MR 3066434
[5] Bravo, J.J., Luca, F.: Repdigits in $k$-Lucas sequences. Proc. Indian Acad. Sci. Math. Sci. 124 (2) (2014), 141–154. DOI 10.1007/s12044-014-0174-7 | MR 3218885
[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. MR 1645552 | Zbl 0911.11018
[7] Faye, B., Luca, F.: Pell and Pell-Lucas numbers with only one distinct digits. Ann. of Math. 45 (2015), 55–60. MR 3438812
[8] Kiliç, E.: The Binet formula, sums and representations of generalized Fibonacci $p$-numbers. European J. Combin. 29 (2008), 701–711. DOI 10.1016/j.ejc.2007.03.004 | MR 2397350
[9] Kiliç, E.: On the usual Fibonacci and generalized order$-k$ Pell numbers. Ars Combin 109 (2013), 391–403. MR 2426404
[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. DOI 10.11650/twjm/1500404581 | MR 2275152
[11] Koshy, T.: Fibonacci and Lucas Numbers with Applications. Pure and Applied Mathematics, Wiley-Interscience Publications, New York, 2001. MR 1855020
[12] Luca, F.: Fibonacci and Lucas numbers with only one distinct digit. Port. Math. 57 (2) (2000), 243–254. MR 1759818 | Zbl 0958.11007
[13] Marques, D.: On $k$-generalized Fibonacci numbers with only one distinct digit. Util. Math. 98 (2015), 23–31. MR 3410879
[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. MR 1817252
[15] Normenyo, B., Luca, F., Togbé, A.: Repdigits as sums of three Pell numbers. Period. Math. Hungarica 77 (2) (2018), 318–328. DOI 10.1007/s10998-018-0247-y | MR 3866634
[16] Normenyo, B., Luca, F., Togbé, A.: Repdigits as sums of four Pell numbers. Bol. Soc. Mat. Mex. (3) 25 (2) (2019), 249–266. DOI 10.1007/s40590-018-0202-1 | MR 3964309
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