CONSTRUCTION OF A SPECIAL INTEGER TRIPLET-I

: This paper is concerned with an interesting Diophantine problem on triplet. A search is made on finding three non-zero distinct integers namely a,b,c such that each of the expressions b a 2  , c a 2  is a perfect square and c b  is twice a cubical integer.Infinitely may such

ABSTRACT: This paper is concerned with an interesting Diophantine problem on triplet. A search is made on finding three non-zero distinct integers namely a,b,c such that each of the expressions b a 2  , c a 2  is a perfect square and c b  is twice a cubical integer.Infinitely may such triplets are obtained.

INTRODUCTION:
Number theory is that branch of Mathematics which deals with properties of the natural numbers 1,2,3,…. also called the positive numbers. These numbers together with the negative numbers and zero form the set of integers. Properties of these integers have been studied since antiquity. Number theory is an art enjoyable and pleasing to everybody. It has fascinated and inspired both amatures and mathematicians alike. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. Certain Diophantine problems come from physical problems or from immediate mathematical generalizations and others come from geometry in a variety of ways. Certain Diophantine problems are neither trivial nor difficult to analyse {AndreWeil [1], Bibhotibhushan Batta and Avadhesh Narayanan [2] ,Boyer.C.B., [3],Dickson.L.E., [4], Davenport , Harold [5],Johnstilwell [6],James Matteson , M.D [7],Titu andreescu , Dorin Andrica [8] such that the sum of the elements in S as well as the sum of any two elements in S is a perfect square. M.A.Gopalan et al., [10], have formulated two interesting triple integer sequences representing harmonic progressions.
A search is made for obtaining two non-zero distinct positive integers, 0 a and 1 a such that i.
is a perfect square. In this communication , we search for different methods of obtaining three non-zero distinct integers a, b, c such that

METHOD OF ANALYSIS:
Let a, b, c be three non-zero distinct integers such that Eliminating a, b, c between the equations (1) -(3), we have (4) is solved through different methods and thus, we obtain infinitely many triples ) , , ( c b a satisfying (1) -(3).

METHOD 1:
Employing the identity Substituting (5) in (1) - (3) , we get the required values of a, b, c to be A few numerical examples are given in Table 1 below:   , b a  is a perfect square.

METHOD 2:
(4) is written as the system of double equations in five different ways as follows: We solve inturn the above five systems of double equations for . The corresponding values of c b a , , satisfying (1) -(3) are exhibited in Table 3 below:   Table 3: Solutions

METHOD 3:
Introduction of the linear transformations  Table 4 below: Using the values of  and  in (1) Table 5 below: In this paper , we have illustrated different methods of obtaining three non-zero distinct integers a, b, c such that b a 2  , c a 2  are respectively perfect square and c b  is 2 times a cubical integer.
As Diophantine problems are rich in variety , one may attempt to construct triples whose elements satisfy special relations among its members.