International Journal of Engineering, Science and Mathematics Vol. 6 Issue 6, October 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijmra.us , Email: editorijmie@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A 107 International Journal of Engineering, Science and Mathematics http://www.ijmra.us , Email: editorijmie@gmail.com On a Polynomial Solutions of a Diophantine Equation Manju Somanath* K. Raja** J. Kannan*** Abstract Let ≔(ݐ) be a polynomial in ݔ. In this paper, we consider the polynomial solutions of Diophantine equation :  2 − 56 2 − 32− 224− 224 = 0. We also obtain some formulae and recurrence relations on the polynomial solution (  ,   ) of . Keywords: Pell equation, Diophantine equation Polynomial solution, Continued fraction expansion. Copyright © 2017 International Journals of Multidisciplinary Research Academy. All rights reserved. Author correspondence: K. Raja, Assistant Professor, Department of Mathematics, National College, Bharathidasan University, Trichy, India. Email: rajakonline@gmail.com 1. Introduction A Diophantine equation is a polynomial equation  ݔ 1 , ݔ 2 , ⋯ , ݔ   =0 where the polynomial  has integral coefficients and one is interested in solutions for which all the unknowns take integer values. For example, ݔ 2 + ݕ 2 = ݖ 2 and ݔ= 3, ݕ= 4, ݖ=5 is one of its infinitely many solutions. Another example is ݔ+ ݕ=1 and all its solutions are given by ݔ= ݐ, ݕ=1 − ݐwhere ݐpasses through all integers. A third example is ݔ 2 +4 ݕ= 3. This Diophantine equation has no solutions, although note that ݔ= 0, ݕ= 3 4 is a solution with rational values for the unknowns. Diophantine equations are rich in variety. Two – variable Diophantine equation have been a subject to extensive research, and their theory constitutes one of the most beautiful, most elaborate part of mathematics, which nevertheless still keeps some of its secrets for the next generation of researchers. In this paper, we investigate positive integral solutions of the Diophantine equation  2 − 56 2 − 32− 224− 224 = 0 which is transformed into a Pell’s equation and is solved by various methods. 2. THE DIOPHANTINE EQUATION  ૛ − ૞૟ ૛ − ૜૛ − ૛૛૝ − ૛૛૝ =  Consider the Diophantine equation :  2 − 56 2 − 32− 224− 224 = 0 (1) to be solved over . It is not easy to solve and find the nature and properties of the solutions of (1). So we apply a linear transformation  to (1) to transfer to a simpler form for which we can determine the integral solutions. Let :   = ݔ+ ℎ  = ݕ+  (2) be the transformation where ℎ, ∈. *Assistant Professor, Department of Mathematics, National College, Trichy, Tamil Nadu, India. **Assistant Professor, Department of Mathematics, National College, Trichy, Tamil Nadu, India. ***Research Scholar, Department of Mathematics, National College, Trichy, Tamil Nadu, India.