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Solutions of the Pell Equations x2 − (a2b2 + 2b)y2 = N when N ∈ {±1,±4}

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Abstract (2. Language): 
Let a and b be natural number and d = a2b2 + 2b. In this paper, by using continued fraction expansion of √d, we find fundamental solution of the equations x2 −dy2 = ±1 and we get all positive integer solutions of the equations x2 − dy2 = ±1 in terms of generalized Fibonacci and Lucas sequences. Moreover, we find all positive integer solutions of the equations x2 − dy2 = ±4 in terms of generalized Fibonacci and Lucas sequences.
629-638

REFERENCES

References: 

[1] Adler, A., Coury, J. E., The Theory of Numbers: A Text and Source Book
of Problems, Jones and Bartlett Publishers, Boston, MA, 1995.
[2] Redmond, D., Number Theory: An Introduction, Markel Dekker, Inc,
1996.
[3] Koninck, J., Mercier, A, 1001 Problems in Classical Number Theory,
American Mathematical Society, 2007.
[4] Robertson, J. P., Solving the generalized Pell equation x2 − Dy2 = N,
http://hometown.aol.com/jpr2718/pell.pdf, May 2003. (Description of
LMM Algorithm for solving Pell’s equation).
[5] Robetson, J. P., On D so that x2−Dy2 represents m and −m and not −1,
Acta Mathematica Academia Paedogogocae Nyiregyhaziensis, 25 (2009),
155-164.
[6] Jones, J. P., Representation of Solutions of Pell Equations Using Lucas
Sequences, Acta Academia Pead. Agr., Sectio Mathematicae 30 (2003),
75-86.
[7] Kalman, D., Mena R., The Fibonacci Numbers exposed, Mathematics
Magazine 76(2003), 167-181.
[8] Keskin, R., Solutions of some quadratic Diophantine equations, Computers
and Mathematics with Applications, 60 (2010), 2225-2230.
[9] McDaniel, W.L., Diophantine Representation of Lucas Sequences, The
Fibonacci Quarterly 33 (1995), 58-63.
[10] Melham, R., Conics Which Characterize Certain Lucas Sequences, The
Fibonacci Quarterly 35 (1997), 248-251.
[11] Jacobson, M. J., Williams, H. C., Solving the Pell Equation, Springer,
2006.
[12] Nagell, T., Introduction to Number Theory, Chelsea Publishing Company,
New York, 1981.
[13] Ribenboim, P., My Numbers, My Friends, Springer-Verlag New York,
Inc., 2000.
[14] Robinowitz, S., Algorithmic Manipulation of Fibonacci Identities, in: Application
of Fibonacci Numbers, vol. 6, Kluwer Academic Pub., Dordrect,
The Netherlands, 1996, pp. 389-408.
638 Merve G¨uney
[15] LeVeque, J. W., Topics in Number Theory, Volume 1 and 2, Dover Publications
2002.
[16] Ismail, M. E. H., One Parameter Generalizations of the Fibonacci and
Lucas Numbers, The Fibonacci Quarterly 46-47 (2009), 167-180.
[17] Zhiwei, S., Singlefold Diophantine Representation of the Sequence u0 =
0, u1 = 1 and un+2 = mun+1 + un, Pure and Applied Logic, Beijing Univ.
Press, Beijing, 97-101, 1992.
[18] Keskin, R., G¨uney, M., Positive Integer Solutions of the Pell Equation
x2 − dy2 = N, d ∈ k2 ± 4, k2 ± 1 and N ∈ ±1,±4(submitted).

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