Formation of Triples Consist Some Special Numbers with Interesting Property

International Research Journal of Engineering and Technology (IRJET)

e-ISSN: 2395-0056

Volume: 04 Issue: 07 | July -2017

p-ISSN: 2395-0072

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FORMATION OF TRIPLES CONSIST SOME SPECIAL NUMBERS WITH INTERESTING PROPERTY V. Pandichelvi1, P. Sivakamasundari2 1Assistant

Professor, Department of Mathematics, UDC, Trichy. Lecturer, Department of Mathematics, BDUCC,Lalgudi, Trichy. ---------------------------------------------------------------------***---------------------------------------------------------------2Guest

Abstract- In this communication, we discover the triple a, b, c  involving some figurate numbers such that the sum

them is a perfect square is given in the following two sections.

of any two of them is a perfect square. Also, we find some interesting relations among the triples.

Section- A

Key words: Diophantine m -tuples, polygonal numbers.

Let

bn  Dec2n2  Gno2n2

INTRODUCTION: Let n be an integer. A set of positive

a1, a2 , a3 ,......am is

integers

said

to

have

an  2CP2n1

the

the property D(n) if ai a j  n is a perfect square for all 1  i  j  m such a sets is called a Diophantine m-tuple. such

sets were studied by Diophantus [1].The set of numbers 1,2,7is Diophantine triple with property D(2).For an

which are equivalent to the following two equations

an  20n 2  30n  12, bn  16n 2  30n  13

Now, we assume that

an  bn    2

Let

extensive review of various articles one may refer [2-12]. In this communication, we find the triple a, b, c  involving centered pentagonal numbers, decagonal numbers, Gnomonic numbers, Kynea numbers and Jacobsthal-lucas numbers such that the sum of any two of them is a perfect square. Also, a few interesting relations among the triples are presented.

cn  be any non-zero integer such that b(n)  cn   2

(1)

a(n)  cn    2

(2)

Subtracting (2) from (1), we get

 2   2  b(n)  an

(3)

Put   A  1,   A in (3), we get

  A  2n 2

Notations:

(4)

Substituting (4) in (2), the values of c are represented by

cn  4n 4  20n 2  30n  12

5n 2  5n  2 Let CPn  be a centered pentagonal number of 2 rank n . t10 ,n  4n  3n be a decagonal number of rank 2

Hence,

20n

n.

j n  2   1 be a Jacobshthal-lucas number of rank n

n

Method of analysis:

|

Impact Factor value: 5.181



 30n  12,16n 2  30n  13,4n 4  20n 2  30n  12 is a

Table-1: Some numerical examples are illustrated below:

n.

The procedure for finding the triple a, b, c  involving some interesting numbers such that the sum of any two of © 2017, IRJET

2

triple in which the sum of any two of them is a perfect square.

n.

Gnon  2n  1 be a Gnomonic number of rank n . K n  2 2n  2 n1  1 be a Kynea number of rank

(5)

|

n

an

bn

cn

a(n)  bn

a(n)  cn

b(n)  cn 

1

62

59

58

112

22

12

2

152

137

88

172

82

72

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