Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Fibonacci and Lucas numbers; exponential Diophantine equation; linear forms in logarithms; Baker's method
Fibonacci and Lucas numbers; exponential Diophantine equation; linear forms in logarithms; Baker's method
Summary:
Let $k\geq 2$ and let $(P_{n}^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence defined by \begin {equation*} P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)} \end {equation*}for $n\geq 2$ with initial conditions \begin {equation*} P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdots =P_{-1}^{(k)}=P_{0}^{(k)}=0,P_{1}^{(k)}=1. \end {equation*}In this study, we handle the equation $P_{n}^{(k)}=y^{m}$ in positive integers $n$, $m$, $y$, $k$ such that $k,y\geq 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_{n}^{(k)}=y^{m}$ with $2\leq y\leq 1000$ has only one solution given by $P_{7}^{(2)}=13^{2}.$
References:
[1] 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
[2] Bravo, J. J., Gómez, C. A., Luca, F.: Powers of two as sums of two $k$-Fibonacci numbers. Miskolc Math. Notes 17 (2016), 85-100. DOI 10.18514/MMN.2016.1505 | MR 3527869 | Zbl 1389.11041
[3] 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
[4] Bravo, J. J., Herrera, J. L., Luca, F.: Common values of generalized Fibonacci and Pell sequences. J. Number Theory 226 (2021), 51-71. DOI 10.1016/j.jnt.2021.03.001 | MR 4239716 | Zbl 1471.11049
[5] 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 07361099
[6] Bravo, J. J., Luca, F.: Powers of two in generalized Fibonacci sequences. Rev. Colomb. Mat. 46 (2012), 67-79. MR 2945671 | Zbl 1353.11020
[7] Bugeaud, Y.: Linear Forms in Logarithms and Applications. IRMA Lectures in Mathematics and Theoretical Physics 28. European Mathematical Society, Zürich (2018). DOI 10.4171/183 | MR 3791777 | Zbl 1394.11001
[8] 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
[9] Cohn, J. H. E.: Square Fibonacci numbers, etc. Fibonacci Q. 2 (1964), 109-113. MR 0161819 | Zbl 0126.07201
[10] Cohn, J. H. E.: Perfect Pell powers. Glasg. Math. J. 38 (1996), 19-20. DOI 10.1017/S0017089500031207 | MR 1373953 | Zbl 0852.11014
[11] Weger, B. M. M. de: Algorithms for Diophantine Equations. CWI Tracts 65. Centrum voor Wiskunde en Informatica, Amsterdam (1989). MR 1026936 | Zbl 0687.10013
[12] Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math., Oxf. II. Ser. 49 (1998), 291-306. DOI 10.1093/qmathj/49.3.291 | MR 1645552 | Zbl 0911.11018
[13] Kiliç, E., Taşci, D.: The generalized Binet formula, representation and sums of the generalized order-$k$ Pell numbers. Taiwanese J. Math. 10 (2006), 1661-1670. DOI 10.11650/twjm/1500404581 | MR 2275152 | Zbl 1123.11005
[14] Ljunggren, W.: Zur Theorie der Gleichung $x^2+1=Dy^4$. Avh. Norske Vid. Akad. Oslo 5 (1942), 1-27 German. MR 0016375 | Zbl 0027.01103
[15] Matveev, E. M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II. Izv. Math. 64 (2000), 1217-1269 translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64 2000 125-180. DOI 10.1070/IM2000v064n06ABEH000314 | MR 1817252 | Zbl 1013.11043
[16] Pethő, A.: The Pell sequence contains only trivial perfect powers. Sets, Graphs and Numbers Colloquia Mathematica Societatis János Bolyai 60. North Holland, Amsterdam (1992), 561-568. MR 1218218 | Zbl 0790.11021
[17] Sanchez, S. G., 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
[18] Şiar, Z., Keskin, R.: On perfect powers in $k$-generalized Pell-Lucas sequence. Available at https://arxiv.org/abs/2209.04190 (2022), 17 pages. MR 4725327
[19] Wu, Z., Zhang, H.: On the reciprocal sums of higher-order sequences. Adv. Difference Equ. 2013 (2013), Article ID 189, 8 pages. DOI 10.1186/1687-1847-2013-189 | MR 3084191 | Zbl 1390.11042
Search
Partner of
