ON THE INTEGRAL SOLUTIONS OF BINARY QUADRATIC DIOPHANTINE EQUATION $ax^{2}-bx=cy^{2}$

Print ISSN: 0972-7752 | Online ISSN: 2582-0850 | Total Downloads : 140

Volume : 17 , Issue : 2
DOI:
Author :
Ashokan Hari Ganesh (Poompuhar College (Autonomous), Melaiyur -- 609107, Nagapattinam (Dt.), Tamil Nadu, INDIA)
Kumaresan Prabhakaran (Annai Vailankanni Arts and Science College, Thanjvur -- 613007, Tamil Nadu, INDIA)

Abstract

In this paper, we show that the Diophantine equation $ax^{2}-bx=cy^{2}$ in positive integers $x, y, a, b, c$ has infinitely many solutions where $ac$ is not a square. We transform the above equation into a Pellian equation to find its infinitely many integer solutions only when $ac$ is not a square. Finally, we present some recurrence relations for $(x, y)$.

Keywords and Phrases

Diophantine Equation, Quadratic Equation, Integral Solutions, Pell's Equation, hyperbola.

A.M.S. subject classification

11DXX, 11D09.

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