Commenced in January 2007
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The Diophantine Equation y2 − 2yx − 3 = 0 and Corresponding Curves over Fp

Authors: Ahmet Tekcan, Arzu Özkoç, Hatice Alkan

Abstract:

In this work, we consider the number of integer solutions of Diophantine equation D : y2 - 2yx - 3 = 0 over Z and also over finite fields Fp for primes p ≥ 5. Later we determine the number of rational points on curves Ep : y2 = Pp(x) = yp 1 + yp 2 over Fp, where y1 and y2 are the roots of D. Also we give a formula for the sum of x- and y-coordinates of all rational points (x, y) on Ep over Fp.

Keywords: Pell equation, Diophantine equation, Quadratic form

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1062334

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References:


[1] E. Barbeau. Pell-s Equation. Springer Verlag, 2003.
[2] J. Buchmann and U. Vollmer. Binary Quadratic Forms: An Algorithmic Approach. Springer-Verlag, Berlin, Heidelberg, 2007.
[3] D.A. Buell. Binary Quadratic Forms, Clasical Theory and Modern Computations. Springer-Verlag, New York, 1989.
[4] H.M. Edwards. 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] D.E. Flath. Introduction to Number Theory. Wiley, 1989.
[6] I. Niven, H.S. Zuckerman and H.L. Montgomery. An Introduction to the Theory of Numbers. Fifth Edition, John Wiley&Sons, Inc., New York, 1991.
[7] A. Tekcan. Pell Equation x2 −Dy2 = 2, II. Bulletin of the Irish Math. Soc. 54(2004), 73-89.
[8] A. Tekcan, O. Bizim and M. Bayraktar. Solving the Pell Equation using the Fundamental Element of the Field Q(ÔêÜΔ). South East Asian Bulletin of Maths 30(2)(2006), 355-366.
[9] A. Tekcan. On the Pell Equation x2−(k2−2)y2 = 2t. Crux Math.with Mathematical Mayhem 33(6)(2007), 361-365.
[10] A. Tekcan. The Pell Equation x2 − Dy2 = ┬▒4. App. Math. Sci. 1(8) (2007), 363-369.
[11] A. Tekcan, B. Gezer, and O. Bizim. On the Integer Solutions of the Pell Equation x2 −dy2 = 2t. Int. Jour. of Math. Sci. 1(3)(2007), 204-208.
[12] A. Tekcan. The Cubic Congruence x3+ax2+bx+c ≡ 0(mod p) and Binary Quadratic Forms F(x, y) = ax2+bxy+cy2. Ars Combinatoria 85(2007), 257-269.
[13] A. Tekcan. The Pell Equation x2 − (k2 − k)y2 = 2t. Int.Journal of Comp. and Mathematical Sciences 2(1)(2008), 5-9.