Browsing Fen-Edebiyat Fakültesi by Author "Karaatlı, Olcay"

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Generalized Fibonacci and Lucas numbers of the form 5x(2)

Keskin, Refik; Karaatlı, Olcay (WORLD SCIENTIFIC PUBL CO PTE LTD, 2015)

Let (U-n(P, Q) and (V-n(P, Q) denote the generalized Fibonacci and Lucas sequences, respectively. In this study, we assume that Q = 1. We determine all indices n such that U-n = 5 square and U-n = 5U(m)square under some ...
GENERALIZED LUCAS NUMBERS OF THE FORM 5kx(2) AND 7kx(2)

Karaatlı, Olcay; Keskin, Refik (KOREAN MATHEMATICAL SOC, 2015)

Generalized Fibonacci and Lucas sequences (U-n) and (V-n) are defined by the recurrence relations Un+1 = PUn +QU(n-1) and Vn+1 = PVn + QV(n-1), n >= 1, with initial conditions U-0 = 0, U-1 = 1 and V-0 = 2, V-1 = P. This ...
Integral points on the elliptic curve y(2) = x(3)+27x-62

Karaatlı, Olcay; Keskin, Refik (SPRINGEROPEN, 2013)

We give a new proof that the elliptic curve y(2) = x(3) + 27x - 62 has only the integral points (x, y) = (2, 0) and (x, y) = (28,844,402, +/- 15,491,585,540) using elementary number theory methods and some properties of ...
On the determination of solutions of simultaneous Pell equations x(2) - (a(2)-1) y(2) = y(2) - pz(2)=1

Keskin, Refik; Karaatlı, Olcay; Zafer Siar; Ummugulsum Ogut (SPRINGER, 2017)

where p is prime and a > 1. Assuming the solutions of the Pell equation x(2) -(a(2) - 1) y(2) = 1 are x = x(m) and y = y(m) with m = 2, we prove that the system (0.1) has solutions only when m = 2 or m = 3. In the case of ...
On the Diophantine equation x(2) - kxy plus y(2)-2(n)=0

Keskin, Refik; Karaatlı, Olcay (SPRINGER HEIDELBERG, 2013)

In this study, we determine when the Diophantine equation x (2)-kxy+y (2)-2 (n) = 0 has an infinite number of positive integer solutions x and y for 0 a (c) 1/2 n a (c) 1/2 10. Moreover, we give all positive integer solutions ...
ON THE DIOPHANTINE EQUATION x(2)-kxy+y(2)+2(n)=0

Keskin, Refik; Karaatlı, Olcay (UNIV MISKOLC INST MATH, 2012)

In this paper, we determine when the equation in the title has an infinite number of positive integer solutions x and y when 0 <= n <= 10. Moreover, we give all the positive integer solutions of the same equation for 0 <= n <= 10.
On the Lucas sequence equations V-n(P, 1) = wkx(2), w epsilon {5,7}

Karaatlı, Olcay (SPRINGER, 2016)

Let P be an odd integer and (V-n) denote the generalized Lucas sequence defined by V-0 = 2, V-1 = P, and Vn+1 = PVn + Vn-1 for n >= 1. In this study, we solve the equations V-n = 5kx(2), V-n = 7kx(2), V-n = 5kx(2) +/- 1, ...
On The Lucas Sequence Equations V-n=7 square and V-n=7V(m)square

Karaatlı, Olcay; Keskin, Refik (SPRINGER, 2018)

Let P be a nonzero integer and let (U-n) and (V-n) denote Lucas sequences of first and second kind defined by U-0 = 0, U-1 = 1; V-0 = 2, V-1 = P; and Un+1 = PUn + Un-1, Vn+1 = PVn + Vn-1 for n >= 1. In this study, when P ...
POSITIVE INTEGER SOLUTIONS OF THE DIOPHANTINE EQUATIONS x(2)-5F(n)(xy)-5(-1)(n)y(2) = +/- 5(r)

Keskin, Refik; Karaatlı, Olcay (UNIV MISKOLC INST MATH, 2013)

In this study, we consider the Diophantine equations given in the title and determine when these equations have positive integer solutions. Moreover, we find all positive integer solutions of them in terms of Fibonacci and ...
The terms of the form 7kx(2) in the generalized Lucas sequence with parameters P and Q

Karaatlı, Olcay (POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN, 2016)