We search for three distinct integers a,b,c such that product of any two from the set added with 1 is a perfect square. Also, we show that the triple can be extended to the quadruple with property D(1)

G Sumathi

M A Gopalan

S Vidhyalakshmi

CiteSeerX

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. 47-50/Gopalan et al.  
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  47 
 
DIOPHANTINE QUADRUPLE INVOLVING JACOBSTHAL LUCAS 
NUMBER AND THABIT-IBN-KURRAH NUMBER WITH PROPERTY 
)1(D  
*M.A. Gopalan, G. Sumathi and S.Vidhyalakshmi 
Department of Mathematics, Shrimathi Indira Gandhi College, Trichy-600002 
*Author for Correspondence  
 
ABSTRACT 
We search for three distinct integers a,b,c such that product of any two from the set added with 1  is a 
perfect square. Also, we show that the triple can be extended to the quadruple with property )1(D   
 
Mathematics Subject Classification: 11D99 
 
Keywords: Diophantine m-Tuples, Thabit-Ibn-Kurrah Number, Jacobsthal Lucas Number, Pellian 
Equation 
 
NOTATIONS   
nTK  : Thabit-ibn-kurrah number of rank n    
nj  : Jacobsthal Lucas number of rank n  
nJ  : Jacobsthal number of rank n  
nKy  : Kynea number of rank n   
  
INTRODUTION 
Let n be an integer. A set of m positive integers ),....,,( 321 maaaa is said to have the property )(nD ,if 
naa ji  is a perfect square for all mji 1 .Such a set is called a diophantine m-tuple or nP set of 
size m.The problem of construction of such set was studied by Diophantus 
Many mathematicians considered the construction of different formulations of Diophantine 
triples, quadruple and quintuples with property )(nD for any arbitrary integer n and also, for any 
linear polynomials in n. In this context, one may refer (Brown, 1985; Bashmakova, 1974; 
Bugeaud et al., 2007; Dujella, 1997; Filipin, 2009; Filipin and Fujita, 2013; Fujita, 2010; 
Gopalan and Srividhya, 2012; Gopalan and Srividhya, 2012; Gopalan and Pandichelvi, 2013; 
Gopalan et al., 2014; Gopalan et al., 2014; Gopalan et al., 2014; Jukic, 2012; Meena et al., 2014; 
Rich, 2012;  Vidhyalakshmi et al., 2014; Andrej) for an extensive review of various problems on 
Diophantine triples, quadruple and quintuples. This paper aims that at constructing Diophantine 
quadruples is constructed where the product of any two members of the quadruple with the addition of   1 
satisfies the required property.  
 
METHOD OF ANALYSIS 
Let 43 2  nja , nTKb 2 be any two integers such that 1ab is a perfect square. 
Let )(nCs be any non-zero integer such that  
 
2
2 1)()43( ssn nCj      (1) 
 
2
2 1)()2( ssn nCTK      (2) 
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. 47-50/Gopalan et al.  
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  48 
 
Eliminating )(nCs between (1) and (2), we get  
 2)43()2( 22
2
2  snsn jTK    (3) 
Introducing the linear transformations 
 





snss
snss
TTKX
TjX
)2(
)43(
2
2


      (4) 
in (3),we get  
 1)2)(43( 222
2  snns TTKjX   (5) 
This is a well known pellian equation,whose general solution is given by 
   
   
   






















1
2
1
2
1
2
1
2
11
2
1
11
2
1
s
n
s
ns
s
n
s
ns
DTKDTK
D
T
DTKDTKX
  (6) 
where .......2,1,0,1s  
 )2)(43( 22  nn TKjD  
Taking 0s  in (6) and using (1),we get 
 1212)( 20  njnC  
Note that ))(,,( 0 nCba is the Diophantine triple with property )1(D  
Now, substituting 1s  in (6) we have 
 






22
14)(2
21
2
2
21
n
nn
TKT
TKTKX
       (7) 
and using (1),we see that  
 1212)1(16)( 2
3
21  nn jTKnC      
Thus, we obtain  
)1212)1(16,1212,2,43( 2
3
2222  nnnnn jTKjTKj  as a diophantine quadruple with the 
property )1(D  
Some numerical examples are presented below: 
 
Diophantine quadruple with property D(1) 
n
 
))(),(,,( 10 nCnCba
 
1 
2 
3 
(11,13,48,27600) 
(47,49,192,1769280) 
191,193,768,113245440)  
 
The repetition of the above process leads to the generation of many diophantine quadruples with the 
property D(1) 
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. 47-50/Gopalan et al.  
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  49 
 
A Few Interesting Properties Observed are As Follows 
1.
 
)4(mod0)(4)(0  nbnC  
2. 2)()(  nanb   
3.
 
)4(mod0)(4)()(0  nanbnC   
4.
 
)2(mod0)(6)()( 02  nCKynbna n   
5. Each of the following is a nasty number 
(a)
 
 )()(18 01 nCnC   
(b)  nJnCnbna 20 54)()()(3   
6.
 
 )()(4 01 nCnC  is a cubic integer  
7.
 
)10(mod036))((
12
3
)()()( 2
3
01  nJnCnCnbna
  
CONCLUSION 
In the construction of the Diophantine quadruples, we have assumed the product ab added with 1 is a 
perfect square. One may search for Diophantine quadruples consisting of special numbers with property 
D(n),D(n
2
 )and so on. 
 
ACKNOWLEDGMENT 
The financial support from the UGC, New Delhi (F-MRP-5123/14(SERO/UGC) dated March 2014) for a 
part of this work is gratefully acknowledged
 
 
REFERENCES 
Andrej Dujella (No Date).  Diophantine m-tuples. Available: web.math.pmf.unizg.hr/~duje/dtuples.html 
Bashmakova IG (1974). Diophantus of Alexandria, Arithmetics and the Book of Polygonal Numbers 
edited by Bashmakova IG (Nauka, Moscow). 
Brown E (1985). Sets in which kxy  is always a square. Mathematics of Computation 45 613-620. 
Bugeaud Y Dujella A and Mignotte (2007). On the family of Diophantine triples,
 
)416,1,1( 3 kkkk  . Glasgow Mathematical Journal 49 333- 344. 
Dujella A (1997). On Diophantine Quintuple. Acta Arithmetica 81 69-79. 
Filipin A (2009). An irregular D(4) quadruple cannot be extended to a quintuple. Acta Arithmetica 136 
167-176 
Filipin A and Fujita Y (2013). The number of Diophantine Quintuples II. Publicationes Mathematicae 
Debrecen 82 293-308. 
Fujita Y (2010). The number of Diophantine Quintuples. Glasnik Matematicki Series III 45 15-29. 
Gopalan M AVidhyalakshmi.S Premalatha E and Presenna R (2014). On the extendibility of 2 tuple 
to 4 tuple with property )4(D . Bulletin of Mathematical Sciences & Applications 3(2) 100-104 . 
Gopalan MA and Pandichelvi V (2013). On the extendibility of the Diophantine triple involving 
Jacobsthal numbers )32,3,( 121221212   nnnnn JJJJJ . Indian Journal of Mathematical 
and Computing Applications 5(2) 83-85. 
Gopalan MA and Srividhya G (2012). Some non-extentable 5P sets. Diophantus Journal of 
Mathematics 1(1) 19-22. 
Gopalan MA and Srividhya G (2012). Two special Diophatine Triples. Diophantus Journal of 
Mathematics 1(1) 23-27. 
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. 47-50/Gopalan et al.  
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  50 
 
Gopalan MA, Mallika S and Vidhyalakshmi S (2014). On the extendibility of the Gaussian 
Diophantine 2-tuple to Gaussian Diophantine Quadruple with property )1(D . International Journal of 
Innovation in Science and Mathematics 2(3) 310-311. 
Gopalan MA, Sumathi G and Vidhyalakshmi S (2014). Special Dio-quadruple involving jacobsthal 
and Jacobsthal lucas number with the Property )1( 2 kD . International Journal of Mathematical 
Sciences and Engineering Applications 8(3) 221-225. 
Jukic Matic Lj (2012). Non-existence of certain Diophantine Quadruples in rings of integers of pure 
cubic fields. Proceedings of the Japan Academy, Series A, Mathematical Sciences 88 163-167. 
Meena K, Vidhyalakshmi S, Gopalan MA, Akila G and Presenna R (2014). Formation of Special 
Diophantine Quadruples with property 
2)6( kpqD . The International Journal of Science and 
Technoledge 2(2) 11-14. 
Rich W (2012). Regular Diophantine m-tuple and their extension. PhD Thesis, Central Michigan 
University. 
Vidhyalakshmi S, Gopalan MA and Lakshmi K (2014). Gaussian Diophantine Quadruples with 
property )1(D . IOSR Journal of Mathematics 10(3) Ver II 12-14.  


DIOPHANTINE QUADRUPLE INVOLVING JACOBSTHAL LUCAS NUMBER AND THABIT-IBN-KURRAH NUMBER WITH PROPERTY D(1)

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=AE7EB3089F2919B0AF81919D04E36ADD?doi=10.1.1.1045.1458&rep=rep1&type=pdf

DIOPHANTINE QUADRUPLE INVOLVING JACOBSTHAL LUCAS NUMBER AND THABIT-IBN-KURRAH NUMBER WITH PROPERTY D(1)

Abstract

Similar works

Full text

Available Versions

CiteSeerX