This paper concerns with the study of constructing a special family of 3-tuples (a,b,c) such that the product of any two elements of the set added with their sum is a Perfect square

E Premalatha

M A Gopalan

R Presenna

S Vidhyalakshmi

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International Journal of Innovative Research and Review ISSN: 2347 – 4424 (Online) 
An Online International Journal Available at http://www.cibtech.org/jirr.htm 
2014 Vol. 2 (2) April-June, pp. 20-22/Gopalan et al.  
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  20 
 
ONE INTERESTING FAMILY OF 3-TUPLE 
M.A. Gopalan1, S. Vidhyalakshmi1, *E. Premalatha2 and R. Presenna3
 
1Department of Mathematics, SIGC, Trichy-620002, Tamilnadu 
2Department of Mathematics, National College, Trichy-62000, Tamilnadu 
3Department of Mathematics, SIGC, Trichy-620002, Tamilnadu 
*Author for Correspondence 
 
ABSTRACT  
This paper concerns with the study of constructing a special family of 3-tuples (a,b,c) such that the 
product of any two elements of the set added with their sum is a Perfect square. 
  
Keywords: Diophantine triple 
 
2010 Mathematics Subject Classification: 11D99 
 
INTRODUCTION  
The Problem of constructing the sets with property that product of any two of its distinct elements is one 
less than a square has a very long history and such sets were studied by Diophantus (Bashmakova, 1974). 
A set of positive integers }......,{ 21 maaa  is said to have the property D(n),  }0{Zn , if naa ji   , a 
perfect square for all  mji 1  and such a set is called a Diophantine m-tuples with property D(n). 
Many mathematicians considered the Construction of different formulations of Diophantine triples with 
property D(n) for any arbitrary integer n and also, for any linear polynomials in n. In this context, one 
may refer (Thamotherampillai, 1980; Brown, 1985; Gupta and Singh, 1985; Beardon and Deshpande, 
2002; Deshpande, 2002; Deshpande, 2003; Bugeaud et al., 2007; Liqun, 2007; Fujita, 2008; Filipin et al., 
2012; Gopalan and Pandichelvi, 2011; Fujita and Togbe, 2011; Gopalan and Srividhya, 2012; Gopalan 
and Srividhya, 2012; Filipin, 2005; Gopalan et al., 2014; Gopalan et al., 2014 and Pandichelvi, 2011) for 
an extensive review of various problem on Diophantine triples. In (Meena et al., Gopalan et al., and 
Gopalan et al.,), special mention is provided because it differs from the earlier one. This paper aims at 
constructing an interesting family of 3-tuples different from the earlier one. The interesting triple is 
constructed where the product of any two members of the triple with addition of the same members is a 
perfect square.  
 
MATERIALS AND METHODS 
Let kkba  2)1(, 222   be such that 
22 ])1[(   kbaab  
Let c be any non zero integer such that  
22222
222
2)1(]12)1[(
)1(
qkkckkcbbc
pccaac




    (1) 
Using some algebra, 
kkqpkk  2)1()1(]12)1[( 22222222     (2) 
Introducing the linear transformations  
TkkXq
TXp
]12)1[(
)1(
22
2




       (3) 
in (2), we have 1]1)1(2)1[( 2222222  TkkX     (4) 
International Journal of Innovative Research and Review ISSN: 2347 – 4424 (Online) 
An Online International Journal Available at http://www.cibtech.org/jirr.htm 
2014 Vol. 2 (2) April-June, pp. 20-22/Gopalan et al.  
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  21 
 
which is in the form of a Pell equation. 
Let   kXT )1(,1 200  be the initial solutions of (4). 
From (3),   )1)(1( 2 kp  
From (1), )1(2)1)(1( 22  kkc   
Hence ))1(2)1)(1(,2)1(,( 22222  kkkk   is the interesting 3-tuple satisfying the 
required property. Repeating the above process, one can generate many 3-tuples satisfying the required 
property. For illustration, a few generated triples are given below.  
 
)42445,420,165()420,165,45(),165,45,4(
),7211050,284218,9168(),284218,9168,462(
),9168,462,22()),2)(1(2,22,1(
),)2()1(4)1(4),1(2)1)(1(,2)1((
22222222
222222
2222
22222222




kkkkkkandkkkkkkkkkk
kkkkkkkkkkkk
kkkkkkkkkk
kkkkk 
Some numerical examples are presented below 
(1, 12, 24), (4, 9, 28), (52, 208, 313), (84, 177, 508), (177, 304, 948) and (264, 456, 1417) 
Remark -1: 
Replacing k by a Gaussian integer a+ib in each of the above triples, it is noted that each resulting triple is 
a Gaussian triple satisfying the required property. 
Eg: (1, 10+10i, 22+14i), (-4+12i, 4+20i, 1+64i), (4, 13+52i, 44+72i), (-4+6i, 7+16i, 4+44i) and (25+32i, 
70+78i, 182+210i). 
Remark -2: 
In a similar manner, one can generate many special 3-tuples such that the product of any two member of 
the set minus the sum of the same members is a perfect square. For illustration, a few examples are given 
below. 
(2, 27, 38), (6, 11, 30), (11, 18, 54), (27, 86, 206), (6, 59, 98), (147, 206, 698), (6, 14, 35) and (66, 107, 
338) 
(3, 4+6i, 12+10i), (12+10i, 24+14i, 67+48i), (2, 7+12i, 14+16i) and (-2+6i, 9+16i, 6+44i) 
 
CONCLUSION 
To conclude, one may search for other triples consisting of other forms of special numbers, namely 
polygonal and centered polygonal numbers. 
 
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International Journal of Innovative Research and Review ISSN: 2347 – 4424 (Online) 
An Online International Journal Available at http://www.cibtech.org/jirr.htm 
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Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  22 
 
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ONE INTERESTING FAMILY OF 3-TUPLE

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ONE INTERESTING FAMILY OF 3-TUPLE

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