On the Lucas sequence equations $$V_{n}(P,1)=wkx^{2},$$ V
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On the Lucas sequence equations Vn ( P, 1) = wkx 2 , w ∈ {5, 7} Olcay Karaatlı1
© Akadémiai Kiadó, Budapest, Hungary 2016
Abstract Let P be an odd integer and (Vn ) denote the generalized Lucas sequence defined by V0 = 2, V1 = P, and Vn+1 = P Vn + Vn−1 for n ≥ 1. In this study, we solve the equations Vn = 5kx 2 , Vn = 7kx 2 , Vn = 5kx 2 ± 1, and Vn = 7kx 2 ± 1 when k|P with k > 1. Moreover, applying some of the results, we obtain complete solutions to the equations Vn = σ x 2 , σ ∈ {15, 21, 35}. Keywords Generalized Fibonacci numbers · Generalized Lucas numbers · Congruences · Jacobi symbol Mathematics Subject Classification
11B37 · 11B39 · 11B50
1 Introduction Let P, Q be nonzero integers, α, β the roots of the polynomial X 2 − P X − Q. Generalized Fibonacci and Lucas sequences with parameters (P, Q) are defined as follows: U0 = 0, U1 = 1, Un+1 = PUn + QUn−1 , V0 = 2, V1 = P, Vn+1 = P Vn + QVn−1 , for all n ≥ 1. We denote these sequences by U = U (P, Q) and V = V (P, Q); if required we use also the notations Un (P, Q) = Un , Vn (P, Q) = Vn . It is convenient to extend the generalized Fibonacci and Lucas sequences also for negative indices: Un Vn U−n = − , V−n = (−Q)n (−Q)n for all n ≥ 1. With this definition, the first two relations hold for all integers n.
B 1
Olcay Karaatlı [email protected] Faculty of Arts and Science, Sakarya University, 54187 Adapazarı, Sakarya, Turkey
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O. Karaatlı
We note also: −1
Un (−P, Q) = (−1)n Un (P, Q), Vn (−P, Q) = (−1)n Vn (P, Q). So, it will be assumed that P ≥ 1. Binet’s formulas express the numbers Un , Vn in terms of α, β : αn − β n Un = , Vn = α n + β n , α−β where α=
P+
P 2 + 4Q P− , β= 2
P 2 + 4Q . 2
Special cases of the sequences (Un ) and (Vn ) are known. For example, the generalized Fibonacci sequence (Un (1, 1)) consists of the familiar Fibonacci numbers, whereas its companion (Vn (1, 1)) gives so called Lucas numbers. When P = 2 and Q = 1; (Un ) = (Pn ) and (Vn ) = (Q n ) are the familiar sequences of Pell and Pell–Lucas numbers. For more information about generalized Fibonacci and Lucas sequences see [5,11–13]. Generalized Fibonacci and Lucas numbers of the form kx 2 have been investigated by many authors and progress in determining the square or k times a square terms of Un and Vn has been made in certain special cases. Interested readers can consult [9,17] for a brief history of this subject. In [14], the authors, applying only congruence properties of sequences, determined all indices n such that Un = x 2 , Un = 2x 2 , Vn = x 2 , and Vn = 2x 2 for all odd relatively prime values of P and Q. Furthermore, the same authors [15] solved Vn = kx 2 under some assumptions on k. In [3], when P is odd, Cohn solved Vn = P x 2 and Vn = 2P x 2 with Q = ±1. In [17], the authors determined all indices n such that Vn (P, 1) = kx 2 when k|P and P is odd, where k is a squarefree positive divisor of P. Keskin [10] found the values of n for which Vn (P, −1) is of the forms kx 2 , 2kx 2 , kx 2 ± 1, and 2kx 2 ± 1 with k|P and k > 1. After, Karaatlı and Keski
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