GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx2 AND wx2 ∓ 1

Title & Authors
GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx2 AND wx2 ∓ 1
Keskin, Refik;

Abstract
Let $\small{P{\geq}3}$ be an integer and let ($\small{U_n}$) and ($\small{V_n}$) denote generalized Fibonacci and Lucas sequences defined by $\small{U_0=0}$, $\small{U_1=1}$; $\small{V_0= 2}$, $\small{V_1=P}$, and $\small{U_{n+1}=PU_n-U_{n-1}}$, $\small{V_{n+1}=PV_n-V_{n-1}}$ for $\small{n{\geq}1}$. In this study, when P is odd, we solve the equations $\small{V_n=kx^2}$ and $\small{V_n=2kx^2}$ with k | P and k > 1. Then, when k | P and k > 1, we solve some other equations such as $\small{U_n=kx^2}$, $\small{U_n=2kx^2}$, $\small{U_n=3kx^2}$, $\small{V_n=kx^2{\mp}1}$, $\small{V_n=2kx^2{\mp}1}$, and $\small{U_n=kx^2{\mp}1}$. Moreover, when P is odd, we solve the equations $\small{V_n=wx^2+1}$ and $\small{V_n=wx^2-1}$ for w = 2, 3, 6. After that, we solve some Diophantine equations.
Keywords
generalized Fibonacci numbers;generalized Lucas numbers;congruences;Diophantine equation;
Language
English
Cited by
1.
GENERALIZED LUCAS NUMBERS OF THE FORM 5kx2 AND 7kx2,;;

대한수학회보, 2015. vol.52. 5, pp.1467-1480
1.
On the equation V n = w x 2 ∓ 1, Arab Journal of Mathematical Sciences, 2017, 23, 2, 148
2.
GENERALIZED LUCAS NUMBERS OF THE FORM 5kx2AND 7kx2, Bulletin of the Korean Mathematical Society, 2015, 52, 5, 1467
3.
Generalized Fibonacci numbers of the form $$wx^{2}+1$$ w x 2 + 1, Periodica Mathematica Hungarica, 2016, 73, 2, 165
4.
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
References
1.
R. T. Bumby, The diophantine equation $3x^4-2y^2=1$, Math. Scand. 21 (1967), 144-148.

2.
J. H. E. Cohn, Squares in some recurrent sequences, Pacific J. Math. 41 (1972), 631-646.

3.
J. P. Jones, Representation of solutions of Pell equations using Lucas sequences, Acta Academia Pead. Agr., Sectio Mathematicae 30 (2003), 75-86.

4.
D. Kalman and R. Mena, The Fibonacci numbersexposed, Math. Mag. 76 (2003), no. 3, 167-181.

5.
R. Keskin, Solutions of some quadratics diophantine equations, Comput. Math. Appl. 60 (2010), no. 8, 2225-2230.

6.
R. Keskin and Z. Siar, Positive integer solutions of some diophantine equations in terms of integer sequences (submitted).

7.
W. L.McDaniel, The g.c.d. in Lucas sequences and Lehmer number sequences, Fibonacci Quart. 29 (1991), no. 1, 24-30.

8.
W. L.McDaniel, Diophantine representation of Lucas sequences, Fibonacci Quart. 33 (1995), no. 1, 58-63.

9.
R. Melham, Conics which characterize certain Lucas sequences, Fibonacci Quart. 35 (1997), no. 3, 248-251.

10.
J. B. Muskat, Generalized Fibonacci and Lucas sequences and rootfinding methods, Math. Comp. 61 (1993), no. 203, 365-372.

11.
S. Rabinowitz, Algorithmic manipulation of Fibonacci identities, Applications of Fi-bonacci numbers, Vol. 6 (Pullman, WA, 1994), 389-408, Kluwer Acad. Publ., Dordrecht, 1996.

12.
P. Ribenboim, My Numbers, My Friends, Springer-Verlag New York, Inc., 2000.

13.
P. Ribenboim and W. L. McDaniel, The square terms in Lucas sequences, J. Number Theory 58 (1996), no. 1, 104-123.

14.
P. Ribenboim and W. L. McDaniel, Squares in Lucas sequences having an even first parameter, Colloq. Math. 78 (1998), no. 1, 29-34.

15.
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, de Gruyter, Berlin, 2001.

16.
Z. Siar and R. Keskin, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 42 (2013), no. 3, 211-222.

17.
Z. Siar and R. Keskin, The square terms in generalized Lucas sequence, Mathematika 60 (2014), 85-100.