Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32128
The Pell Equation x2 − Py2 = Q

Authors: Ahmet Tekcan, Arzu Özkoç, Canan Kocapınar, Hatice Alkan


Let p be a prime number such that p ≡ 1(mod 4), say p = 1+4k for a positive integer k. Let P = 2k + 1 and Q = k2. In this paper, we consider the integer solutions of the Pell equation x2-Py2 = Q over Z and also over finite fields Fp. Also we deduce some relations on the integer solutions (xn, yn) of it.

Keywords: Pell equation, solutions of Pell equation.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1953


[1] Arya S.P. On the Brahmagupta-Bhaskara Equation. Math. Ed. 8(1) (1991), 23-27.
[2] Baltus C. Continued Fractions and the Pell Equations:The work of Euler and Lagrange. Comm. Anal. Theory Contin. Fractions 3(1994), 4-31.
[3] Barbeau E. Pell-s Equation. Springer Verlag, 2003.
[4] Edwards, H.M. Fermat-s Last Theorem. A Genetic Introduction to Algebraic Number Theory. Corrected reprint of the 1977 original. Graduate Texts in Mathematics, 50. Springer-Verlag, New York, 1996.
[5] Kaplan P. and Williams K.S. Pell-s Equations x2−my2 = −1,−4 and Continued Fractions. Journal of Number Theory. 23(1986), 169-182.
[6] Lenstra H.W. Solving The Pell Equation. Notices of the AMS. 49(2) (2002), 182-192.
[7] Matthews, K. The Diophantine Equation x2 − Dy2 = N, D > 0. Expositiones Math.18 (2000), 323-331.
[8] Mollin R.A., Poorten A.J. and Williams H.C. Halfway to a Solution of x2 −Dy2 = −3. Journal de Theorie des Nombres Bordeaux, 6(1994), 421-457.
[9] Niven I., Zuckerman H.S. and Montgomery H.L. An Introduction to the Theory of Numbers. Fifth Edition, John Wiley&Sons, Inc., New York, 1991.
[10] Stevenhagen P. A Density Conjecture for the Negative Pell Equation. Computational Algebra and Number Theory, Math. Appl. 325(1992), 187-200.
[11] Tekcan A. Pell Equation x2 − Dy2 = 2, II. Bulletin of the Irish Mathematical Society 54 (2004), 73-89.
[12] Tekcan A., Bizim O. and Bayraktar M. Solving the Pell Equation Using the Fundamental Element of the Field Q(ÔêÜΔ). South East Asian Bull. of Maths. 30(2006), 355-366.
[13] Tekcan A. The Pell Equation x2 − Dy2 = ┬▒4. Applied Mathematical Sciences, 1(8)(2007), 363-369.
[14] Tekcan A., Gezer, B. and Bizim, O. On the Integer Solutions of the Pell Equation x2 − dy2 = 2t. Int. Journal of Computational and Mathematical Sciences 1(3)(2007), 204-208.
[15] Tekcan A. On the Pell Equation x2 − (k2 − 2)y2 = 2t. Crux Mathematicorum with Mathematical Mayhem 33(6)(2007), 361-365.
[16] Tekcan A. The Pell Equation x2−(k2−k)y2 = 2t. International Jour. of Comp. and Mathematical Sci. 2(1)(2008), 5-9.
[17] Tekcan A. and Bizim O. The Pell Equation x2 + xy − ky2 = ┬▒1. Global Journal of Pure and Applied Mathematics 4(2)(2008),
[18] Tekcan A., O┬¿ zkoc┬© A. and Alkan H. The Diophantine Equation y2 − 2yx − 3 = 0 and Corresponding Curves over Fp. International Jour. of Math.and Statis. Sci. 1 (2)(2009), 66-69.
[19] Tekcan A. and O┬¿ zkoc┬© A. The Diophantine Equation x2 −(t2 +t)y2 − (4t + 2)x + (4t2 + 4t)y = 0. Revista Matem'atica Complutense 23(1)(2010), 251-260.
[20] Tekcan A. The Number of Solutions of Pell Equations x2 − ky2 = N and x2 + xy − ky2 = N over Fp. Accepted for publication to Ars Combinatoria.
[21] O┬¿ zkoc┬© A. and Tekcan A. Quadratic Diophantine Equation x2 − (t2 − t)y2 −(4t−2)x+(4t2 −4t)y = 0. Bull. of the Malaysian Math. Sci. Soc. 33(2)(2010), 273-280.
[22] O┬¿ zkoc┬© A., Kocap─▒nar C. and Tekcan A. Some Algebraic Identities on the Sequence A = An(P,Q) with Parameters P and Q. Submitted for publication.