GENERALIZED LUCAS NUMBERS OF THE FORM 5kx2 AND 7kx2
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Generalized Fibonacci and Lucas sequences (Un) and (Vn) are defined by the recurrence relations Un+1 = PUn+QUn−1 and Vn+1 = PVn +QVn−1, n ≥ 1, with initial conditions U0 = 0, U1 = 1 and V0 = 2, V1 = P. This paper deals with Fibonacci and Lucas numbers of the form Un(P,Q) and Vn(P,Q) with the special consideration that P ≥ 3 is odd and Q = −1. Under these consideration, we solve the equations Vn = 5kx2, Vn = 7kx2, Vn = 5kx2±1, and Vn = 7kx2±1 when k | P with k > 1. Moreover, we solve the equations Vn = 5x2 ±1 and Vn = 7x2 ± 1.
Abstract
Generalized Fibonacci and Lucas sequences (Un) and (Vn) are defined by the recurrence relations Un+1 = PUn+QUn−1 and Vn+1 = PVn +QVn−1, n ≥ 1, with initial conditions U0 = 0, U1 = 1 and V0 = 2, V1 = P. This paper deals with Fibonacci and Lucas numbers of the form Un(P,Q) and Vn(P,Q) with the special consideration that P ≥ 3 is odd and Q = −1. Under these consideration, we solve the equations Vn = 5kx2, Vn = 7kx2, Vn = 5kx2±1, and Vn = 7kx2±1 when k | P with k > 1. Moreover, we solve the equations Vn = 5x2 ±1 and Vn = 7x2 ± 1.
참고문헌 (14)
- J. H. E. Cohn, "Squares in some recurrent sequences", Pacific J. Math, 41, pp.631~646 (1972)
- B. Demirt¨urk, "Integer solutions of some Diophantine equations via Fi-bonacci and Lucas numbers", J. Integer Seq, 12(8), pp.14~ (2009)
- D. Kalman, "The Fibonacci numbers-exposed", Math. Mag, 76(3), pp.167~181 (2003)
- Refik Keskin, "GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx 2 AND wx 2 ∓ 1", 대한수학회보, 51(4), pp.1041~1054 (2014), 대한수학회
- R. Keskin, "Generalized Fibonacci and Lucas numbers of the form 5x2", Int. J. Number Theory, 11(3), pp.931~944 (2015)
- W. L. McDaniel, "Diophantine Representation of Lucas Sequences", Fibonacci Quart, 33(1), pp.59~63 (1995)
- R. Melham, "Conics which characterize certain Lucas sequences", Fibonacci Quart, 35(3), pp.248~251 (1997)
- J. B. Muskat, "Generalized Fibonacci and Lucas sequences and rootfinding methods", Math. Comp, 61(203), pp.365~372 (1993)
- S. Rabinowitz, "Algorithmic manipulation of Fibonacci identities", Appl. Fibonacci Numbers, 6, pp.389~408 (1996)
- P. Ribenboim, "My Numbers, My Friends" (2000), Springer-Verlag
- P. Ribenboim, "The square terms in Lucas sequences", J. Number Theory, 58(1), pp.104~123 (1996)
- P. Ribenboim, "On Lucas sequence terms of the form kx2, Number Theory", Proceedings of the Turku symposium on Number Theory in memory of Kustaa Inkeri, pp.293~303 (2001), Walter de Gruyter
- Z. S¸iar, "Some new identities concerning generalized Fibonacci and Lucas numbers", Hacet. J. Math. Stat, 42(3), pp.211~222 (2013)
- Z. S¸iar, "The square terms in generalized Fibonacci sequence", Mathematika, 60(1), pp.85~100 (2014)