On The Ternary Quadratic Diophantine Equation

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International Journal for Research in Applied Science & Engineering Technology (IJRASET)

1Assistant Professor, 2PG student, PG and Research Department of Mathematics Cauvery College for Women (Autonomous) Trichy-18, India (Affiliated Bharathidasan University)

Abstract: The non-zero unique integer solutions to the quadratic Diophantine equation with three unknowns are examined. We derive integral solutions in four different patterns. A few intriguing relationships between the answers and a few unique polygonal integersare shown.

Keywords: Ternary quadraticequation, integral solutions.

I. INTRODUCTION

There is a wide range of ternary quadratic equations. One might refer to [1-8] for a thorough review of numerous issues. These findings inspired us to look for an endless number of non-zero integral solutions to another intriguing ternary quadratic problem provided by 2 2 2 14 z y xy x    illustrating a cone for figuring out its many non-zero integral points. A few intriguing connections between the solutions are displayed.

II. CONNECTED WORK

 a Pr Pronic number of the rank ‘n’

 a Gno Gnomonic number of rank ‘n’

 n m T , Polygonal number of rank ‘n’ with sides ‘m’

III. METHODOLOGY

The ternary quadratic Diophantine equation to be solved for its non-zero integral solutions is,

(1) Replacement of linear transformations

  x and    y

(1) results in

12 16 z  

We present below different patterns of solving (3) and thus obtain different choices of integer solutions of (1)

A. Pattern: 1

Assume, ) , ( b a z z  = 2 2 12 16 b a

Where a and b are non-zero integers. Substitute (4) in (3) we get,

Equating rational and irrational terms we get,