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Keywords:
Diophantine equation; Narayana's cows sequence; $k$-Pell number; linear form in logarithms; reduction method
Diophantine equation; Narayana's cows sequence; $k$-Pell number; linear form in logarithms; reduction method
Summary:
For any positive integer $k\geq 2$, let $(P_n^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots ,0,1$ ($k$ terms) with the linear recurrence $$ P_{n}^{(k)} = 2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}\quad \text {for}\ n\geq 2. $$ Let $(N_n)_{n\geq 0}$ be Narayana's sequence given by $$ N_0=N_1=N_2=1\quad \text {and}\quad N_{n+3}=N_{n+2}+N_{n}. $$ The purpose of this paper is to determine all $k$-Pell numbers which are sums of two Narayana's numbers. More precisely, we study the Diophantine equation $$ P_p^{(k)}=N_n+N_m $$ in nonnegative integers $k$, $p$, $n$ and $m$.
References:
[1] Allouche, J.-P., Johnson, T.: Narayana's cows and delayed morphisms. 3rd Computer Music Conference JIM96 IRCAM, Paris (1996), 6 pages.
[2] Baker, A., Davenport, H.: The equations $3x^2-2=y^2$ and $8x^2-7=z^2$. Q. J. Math., Oxf. II. Ser. 20 (1969), 129-137. DOI 10.1093/qmath/20.1.129 | MR 0248079 | Zbl 0177.06802
[3] Bhoi, K., Ray, P. K.: Fermat numbers in Narayana's cows sequence. Integers 22 (2022), Article ID A16, 7 pages. MR 4369871 | Zbl 1493.11032
[4] Bravo, J. J., Das, P., Guzmán, S.: Repdigits in Narayana's cows sequence and their consequences. J. Integer Seq. 23 (2020), Article ID 20.8.7, 15 pages. MR 4161574 | Zbl 1477.11031
[5] Bravo, J. J., Herrera, J. L.: Repdigits in generalized Pell sequences. Arch. Math., Brno 56 (2020), 249-262. DOI 10.5817/AM2020-4-249 | MR 4173077 | Zbl 07285963
[6] Bravo, J. J., Herrera, J. L., Luca, F.: On a generalization of the Pell sequence. Math. Bohem. 146 (2021), 199-213. DOI 10.21136/MB.2020.0098-19 | MR 4261368 | Zbl 1499.11049
[7] Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Ann. Math. (2) 163 (2006), 969-1018. DOI 10.4007/annals.2006.163.969 | MR 2215137 | Zbl 1113.11021
[8] Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math., Oxf. II. Ser. 49 (1998), 291-306. MR 1645552 | Zbl 0911.11018
[9] Faye, M. N., Rihane, S. E., Togbé, A.: On repdigits which are sum or differences of two $k$-Pell numbers. Math. Slovaca 73 (2023), 1409-1422. DOI 10.1515/ms-2023-0102 | MR 4678548 | Zbl 7791584
[10] Sanchez, S. Guzmán, Luca, F.: Linear combinations of factorials and $S$-units in a binary recurrence sequence. Ann. Math. Qué. 38 (2014), 169-188. DOI 10.1007/s40316-014-0025-z | MR 3283974 | Zbl 1361.11007
[11] Kiliç, E.: On the usual Fibonacci and generalized order-$k$ Pell numbers. Ars Comb. 88 (2008), 33-45. MR 2426404 | Zbl 1224.11024
[12] Normenyo, B. V., Rihane, S. E., Togbé, A.: Common terms of $k$-Pell numbers and Padovan or Perrin numbers. Arab. J. Math. 12 (2023), 219-232. DOI 10.1007/s40065-022-00407-8 | MR 4552849 | Zbl 1523.11035
[13] Sloane, N. J. A.: The On-Line Encyclopedia of Integer Sequences. Available at https://oeis.org/ (2019). MR 3822822
[14] Wu, Z., Zhang, H.: On the reciprocal sums of higher-order sequences. Adv. Difference Equ. 2013 (2013), Paper ID 189, 8 pages. DOI 10.1186/1687-1847-2013-189 | MR 3084191 | Zbl 1390.11042
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