GENERALIZED LUCAS NUMBERS OF THE FORM 5kx2 AND 7kx2

Title & Authors
GENERALIZED LUCAS NUMBERS OF THE FORM 5kx2 AND 7kx2
KARAATLI, OLCAY; KESKIN, REFIK;

Abstract
Generalized Fibonacci and Lucas sequences ($\small{U_n}$) and ($\small{V_n}$) are defined by the recurrence relations $\small{U_{n+1}=PU_n+QU_{n-1}}$ and $\small{V_{n+1}=PV_n+QV_{n-1}}$, $\small{n{\geq}1}$, with initial conditions $\small{U_0=0}$, $\small{U_1=1}$ and $\small{V_0=2}$, $\small{V_1=P}$. This paper deals with Fibonacci and Lucas numbers of the form $\small{U_n}$(P, Q) and $\small{V_n}$(P, Q) with the special consideration that $\small{P{\geq}3}$ is odd and Q = -1. Under these consideration, we solve the equations $\small{V_n=5kx^2}$, $\small{V_n=7kx^2}$, $\small{V_n=5kx^2{\pm}1}$, and $\small{V_n=7kx^2{\pm}1}$ when $\small{k{\mid}P}$ with k > 1. Moreover, we solve the equations $\small{V_n=5x^2{\pm}1}$ and $\small{V_n=7x^2{\pm}1}$.
Keywords
generalized Fibonacci numbers;generalized Lucas numbers;congruences;
Language
English
Cited by
1.
On the Lucas sequence equations $$V_{n}(P,1)=wkx^{2},$$ V n ( P , 1 ) = w k x 2 , $$w\in \left\{ 5,7\right\}$$ w ∈ 5 , 7, Periodica Mathematica Hungarica, 2016, 73, 1, 73
2.
On the equation V n = w x 2 ∓ 1, Arab Journal of Mathematical Sciences, 2017, 23, 2, 148
3.
Generalized Fibonacci numbers of the form $$wx^{2}+1$$ w x 2 + 1, Periodica Mathematica Hungarica, 2016, 73, 2, 165
References
1.
J. H. E. Cohn, Squares in some recurrent sequences, Pacific J. Math. 41 (1972), 631-646.

2.
B. Demirturk and R. Keskin, Integer solutions of some Diophantine equations via Fibonacci and Lucas numbers, J. Integer Seq. 12 (2009), no. 8, Article 09.8.7, 14 pp.

3.
D. Kalman and R. Mena, The Fibonacci numbers-exposed, Math. Mag. 76 (2003), no. 3, 167-181.

4.
R. Keskin, Generalized Fibonacci and Lucas numbers of the form $wx^2$ and $wx^2$ ${\pm}$ 1, Bull. Korean Math. Soc. 51 (2014), no. 4, 1041-1054.

5.
R. Keskin and O. Karaatli, Generalized Fibonacci and Lucas numbers of the form $5x^2$, Int. J. Number Theory 11 (2015), no. 3, 931-944.

6.
W. L. McDaniel, Diophantine Representation of Lucas Sequences, Fibonacci Quart. 33 (1995), no. 1, 59-63.

7.
R. Melham, Conics which characterize certain Lucas sequences, Fibonacci Quart. 35 (1997), no. 3, 248-251.

8.
J. B. Muskat, Generalized Fibonacci and Lucas sequences and rootfinding methods, Math. Comp. 61 (1993), no. 203, 365-372.

9.
S. Rabinowitz, Algorithmic manipulation of Fibonacci identities, Appl. Fibonacci Numbers 6 (1996), 389-408.

10.
P. Ribenboim, My Numbers, My Friends, Springer-Verlag New York, Inc., 2000.

11.
P. Ribenboim and W. L. McDaniel, The square terms in Lucas sequences, J. Number Theory 58 (1996), no. 1, 104-123.

12.
P. Ribenboim and W. L. McDaniel, On Lucas sequence terms of the form $kx^2$, Number Theory: Proceedings of the Turku symposium on Number Theory in memory of Kustaa Inkeri (Turku, 1999), 293-303, Walter de Gruyter, Berlin, 2001.

13.
Z. Siar and R. Keskin, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 42 (2013), no. 3, 211-222.

14.
Z. Siar and R. Keskin, The square terms in generalized Fibonacci sequence, Mathematika 60 (2014), no. 1, 85-100.