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Fibonacci numbers which are products of three Pell numbers and Pell numbers which are products of three Fibonacci numbers. (English) Zbl 1484.11058

In this work, the authors intend to extend some of the previous works related to Fibonacci and Pell numbers. One major Diophantine equation considered in the previous works was the intersection of the two binary recurrence sequence \(F_n\) (the Fibonacci numbers) and \(P_n\) (the Pell numbers). For which only finite solutions were obtained by Alekseyev. Then, as an extension Fibonacci numbers which are product of two Pell numbers and conversely, the Pell numbers which are product of two Fibonacci numbers were studied. Using the theory of linear forms in logarithms by obtaining the lowers bounds on those linear forms of logarithms and some reduction methods due to De Weger to show all the solutions of the Diophantine equations \(F_n=P_mP_kP_l\) and \(P_n=F_mF_kF_l\), where \(l\leq k \leq m\) that is, the authors obtained all Fibonacci numbers which are product of three Pell numbers and all Pell numbers which are product of three Fibonacci numbers.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11J86 Linear forms in logarithms; Baker’s method
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References:

[1] Alekseyev, MA, On the intersections of Fibonacci, Pell, and Lucas numbers, Integers, 11, 239-259 (2011) · Zbl 1228.11018 · doi:10.1515/integ.2011.021
[2] Ddamulira, M.; Luca, F.; Rakotomalala, M., Fibonacci numbers which are products of two Pell numbers, Fibonacci Q., 54, 1, 11-18 (2016) · Zbl 1400.11040
[3] Dujella, A.; Pethő, A., A generalization of a theorem of baker and davenport, Q. J. Math. Oxf. Ser. (2), 49, 195, 291-306 (1998) · Zbl 0911.11018 · doi:10.1093/qjmath/49.195.291
[4] Matveev, EM, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II, Izv. Math., 64, 6, 1217-1269 (2000) · Zbl 1013.11043 · doi:10.1070/IM2000v064n06ABEH000314
[5] Weger, BMM, Algorithms for Diophantine Equations (1989), Amsterdam: Stichting Mathematisch Centrum, Amsterdam
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