[Maheswari et al., 3(8): August, 2014] ISSN: 2277-9655 Scientific Journal Impact Factor: 3.449 (ISRA), Impact Factor: 1.852 http: // www.ijesrt.com (C)International Journal of Engineering Sciences & Research Technology [193] IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY On The Binary Quadratic Diophantine Equation ࢞ − ࢞࢟ + ࢟ + ࢞ = M.A.Gopalan, S.Vidhyalakshmi, D.Maheswari* Department of Mathematics, Shrimati Indira Gandhi College, India matmahes@gmail.com Abstract The binary quadratic equation ݔଶ − ݕݔ+ ݕଶ +ͷ =ݔͲ representing hyperbola is considered and analysed for its integer points. A few interesting relations satisfied by x and y are exhibited. Keywords: binary quadratic, hyperbola, integer solutions 2010 Mathematics Subject Classification : 11D09. Introduction The binary quadratic equation offers an unlimited field for research because of their variety [1-5]. In this context, one may also refer [6-21]. This communication concerns with yet another interesting binary quadratic equation ݔଶ − ݕݔ+ ݕଶ +ͷ =ݔͲ for determining its infinitely many non-zero integral solutions. Also a few interesting relations among the solutions are presented. Method of analysis The hyperbola under consideration is ݔଶ − ݕݔ+ ݕଶ +ͷ =ݔͲ (1) To start with, it is seen that (1) is satisfied by the following pairs of integers (1, 1), (-5, 0), (36, 6), (-245, -35). However we have other choices of solutions satisfying (1) and they are illustrated below: Choice 1: Considering (1) as a quadratic in x and solving for x, we get =ݔଵ ଶ [ ݕ− ͷ ± √Ͷͷ ݕଶ − Ͳ ݕ+ ʹͷ ] (2) Let ߙଶ = Ͷͷ ݕଶ − Ͳ ݕ+ ʹͷ (3) Substituting =ߙ ଷ and =ݕ+ ଽ (4) in (3), we have ߚଶ = ͷ ଶ − ʹͲ (5) whose least positive solution is ߚ = ͷ, = ͵. Considering the solutions ( ߚ , ) of the pellian equation ߚଶ = ͷ ଶ +ͳ and applying Brahmagupta lemma between ሺ ߚ , ሻ and ( ߚ , ), we have ߚ+ଵ = ହ ଶ + ଷ√ହ ଶ , +ଵ = ଷ ଶ + √ହ ଶ (6) where = ሺͻ + Ͷ√ͷ ሻ +ଵ + ሺͻ − Ͷ√ͷ ሻ +ଵ and = ሺͻ + Ͷ√ͷ ሻ +ଵ − ሺͻ − Ͷ√ͷ ሻ +ଵ , n = 0, 1, 2, …. Substituting (4) and (6) in (2) and taking the positive sign, the corresponding integer solutions to (1) are given by ݔସ+ଵ ܨ=+ ସ√ହ ଽ ܩ+ ଶ ଽ ݕ,ସ+ଵ = ଵ ܨ+ √ହ ଵ ܩ+ ଽ where =ܨሺͻ + Ͷ√ͷ ሻ ସ+ଵ + ሺͻ − Ͷ√ͷ ሻ ସ+ଵ and G= ሺͻ + Ͷ√ͷ ሻ ସ+ଵ − ሺͻ − Ͷ√ͷ ሻ ସ+ଵ , k = 0, 1, 2, 3, …. The recurrence relations satisfied by x and y are given by ݔସ+ଽ = ͳͲ͵ͺʹ ݔସ+ହ − ݔସ+ଵ − ʹ͵ͲͶͲ; ݔଵ = ͵, ݔହ = ͵ͲͻͶ ݕସ+ଽ = ͳͲ͵ͺʹ ݕସ+ହ − ݕସ+ଵ − ͺͲͶͲ; ݕଵ = , ݕହ = ͷͶͳʹͲ Some numerical examples of x and y satisfying (1) are given in the following table: