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On Polynomials Solutions of Quadratic Diophantine Equations
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Advances in Pure Mathematics·Volume 01 (2011)·Pages 155–159·Published 22 July 2011·DOI10.4236/apm.2011.14028
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Abstract
Let <i>P</i>:=<i>P</i>(<i>t</i>) be a polynomial in Z[<i>X</i>]\{0,1} In this paper, we consider the number of polynomial solutions of Diophantine equation <i>E</i>:<i>X</i><sup>2</sup>–(<i>P</i><sup>2</sup>–<i>P</i>)<i>Y</i><sup>2</sup>–(4<i>P</i><sup>2</sup>–2)<i>X</i>+(4<i>P</i><sup>2</sup>–4<i>P</i>)<i>Y</i>=0. We also obtain some formulas and recurrence relations on the polynomial solution (<i>X</i><sub><i>n</i></sub>,<i>Y</i><sub><i>n</i></sub>) of <i>E</i>
KeywordsPolynomial SolutionsPell’s EquationDiophantine Equation
- I. Niven, H. S. Zuckerman and H. L. Montgomery, “An Introduction to the Theory of Numbers,” 5th Edition, Ox-ford University Press, Oxford, 1991.
- A. Tekcan, “The Pell Equation ,” Applied Mathematical Sciences, Vol. 1, No. 8, 2007, pp. 363-369.
- P. Kaplan and K. S Williams, “Pell’s Equation and Continued Fractions,” Journal of Num-ber Theory, Vol. 23, No. 2, 1986, pp. 169-182. doi:10.1016/0022-314X(86)90087-9
- K. Matthews, “The Diophantine Equation ,” Expositiones Mathematicae, Vol. 18, 2000, pp. 323-331.
- R. A. Mollin, A. J Poorten and H. C. Williams, “Halfway to a Solution of ,” Journal de Theorie des Nombres Bordeaux, Vol. 6, No. 2, 1994, pp. 421-457.
- P. Stevenhagen, “A Density Conjecture for the Negative Pell Equation, Computational Algebra and Number Theory,” Math. Appl. Vol. 325, 1992, pp. 187-200.
- A. Chandoul, “The Pell Equation ,” Research Journal of Pure Algebra, Vol. 1, No. 2, 2011, pp. 11-15.
- A. S. Shabani, “The Proof of Two Conjectures Related to Pell’s Equation ,” International Journal of Computational and Mathematical Sciences, Vol. 2, No. 1, 2008, pp. 24-27.
- A. Chandoul, “The Pell Equation ,” Advances in Pure Mathematics, Vol. 1, No. 2, 2011, pp. 16-22. doi:10.4236/apm.2011.12005
- A. Dubickas and J. Steuding, “The Polynomial Pell Equation,” Elemente der Mathematik, Vol. 59, No. 4, 2004, pp. 133-143. doi:10.1007/s00017-004-0214-7
- A. Tekcan, “Quadratic Diophantine Equation , Bulletin of Malay-sian Mathematical Society, Vol. 33, No. 2, 2010, pp. 273-280.
- A. Chandoul, “On Quadratic Diophantine Equation ,” International Mathematical Forum, Vol. 6, No. 36, 2011, pp. 1777-1782.