CONSTRUCTION OF IRRATIONAL GAUSSIAN DIOPHANTINE QUADRUPLES

  • M. A. Gopalan Professor, Department of mathematics, Shrimathi Indira Gandhi College, Trichy, Tamilnadu, INDIA
  • S.Vidhyalakshmi Professor, Department of mathematics, Shrimathi Indira Gandhi College, Trichy, Tamilnadu, INDIA
  • N.Thiruniraiselvi Research Scholar, Department of mathematics, Shrimathi Indira Gandhi College, Trichy, Tamilnadu, INDIA
Keywords: Diophantine quadruple, irrational Diophantine quadruples, Gaussian Diophantine quadruples, Pell equations, Double Diophantine equations, integer solutions

Abstract

Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.

Downloads

Download data is not yet available.

References

I.E.Dickson,History of the numbers,Vol.2 Chelsea,Newyork,1966,513-520.

E.Brown,Sets in which xy+k is always a square,Math.Comp.,1985 ,45,613-620. DOI: https://doi.org/10.2307/2008150

J.Morgado,Generalization of a result of Hoggatt and Bergum on Fibonacci numbers Portugaliae Math,42(1983-1984),441-445.

J.Morgado,Note on a Shannon’s theorem concerning the Fibonacci numbers and Diophantine quadruples,Portugaliae Math,48(1991),429-439.

G.Srividhya, Diophantine quadruples for Fibonacci numbers with property D(1),Indian Journal of Mathematics and Mathematical Sciences,5, (2009), 57-59.

M.A.Gopalan,G.Shrividhya Diophantine quadraples for Fibanacci and Lucas numbers with property D(4) Diophantus J.Math 1, ,(2012), 15-18.

A.DujellaGeneralization of a problem of diophantus,Acta Arith 65,(1993) ,15-27. [Gopalan et al., Vol.1 (Iss.1): April, 2015] ISSN: 2454-1907 Http://www.ijetmr.com©International Journal of Engineering Technologies and Management Research [1-7]

A.DujellaA problem of Diophantus and pell numbers,Application of Fibanacci Numbers,Vol 7( G.E.Bergum,A.N.Philippou,A.F.Horadan eds.) Khowet Dordrecht, ,(1998) , 61-68. DOI: https://doi.org/10.1007/978-94-011-5020-0_9

Y.FujitaThe extensibility of Diophantine pairs [k-l,k+l],J.Number Theory,128, (2008)322-353.

Y.Fujita,A.TogbeUniqueness of the extension of the D(4k2 )-triple [k2 -4,k2 ,4k2 -4],Notes Number Theory,Discrete Math,17, (2011),42-49.

Z.Cerin,On extended Euler quadruples, Atti Semin Mat Fis.Univ Modena Reggio Emilia 58 ,(2011),101-120.

Y.Zhang,Diophantine triples and extendibility of [1,2,5] and [1,5,10],Master Thesis,Central Michigan University ,(2011)..

Lj.Bacic,A.Filipin,The extendibility of D(4)-pairs Math,Commun,18,(2013),447- 456.

K.Meena,S.Vidhyalakshmi,M.A.Gopalan,G.Akila,R.Presenna,(2014)Formation of special Diophatine quadraples with property D(6kpq)2 ,The international journal of Science and Technology 2,11-14.

Andrej Dujella and Florian Luca,Diophantine m-tuples for primes Intern.Math.Research,Notes 47 ,(2005),2913-2940. DOI: https://doi.org/10.1155/IMRN.2005.2913

Gopalan.M.A,Lakshmi,KVidhyalakshmi,S, Gaussian Diophantine quadruples with property D(1),IOSR,Vol 10 (3),version II,1,(2014),2-14. DOI: https://doi.org/10.9790/5728-10321214

Gopalan.M.A,Mallika,S,Vidhyalakshmi.S,On the extendibility of the Gaussian 2- tuple to Gaussian Diophantine quadruple with property D(1),International Journal of Innovation in Science and Mathematics,Vol.2,Issue 3, ,(2014),310-311. DOI: https://doi.org/10.9790/5728-10321214

Gopalan.M.A,Mallika,S,Vidhyalakshmi.S,On Diophantine quadruple with property ( ) 2 D P where P is prime and 1(mod6) 2 P  International Journal of Current Research,Vol.6,Issue.5,(2014), 6810-6813.

K.Meena,S.Vidhyalakshmi,M.A.Gopalan,R.Presenna Sequences of special Dio-

Triples,IJMTT,Vol 10,No.1, (2014),43-46., DOI: https://doi.org/10.1515/george-2014-0009

Gopalan.M.A,Mallika,S,Vidhyalakshmi.S,On Sequences of irrational a. Diophantine quadruples International Journal of Mathematical Sciences andb. Applications,Vol.4,No.1, (2014),128-135.

Gopalan.M.A,Vidhyalakshmi.S,Thiruniraiselvi.N,”On strong Rational Diophantine Quadruples with equal members”, SJPMS, 2014,Vol-1,Issue-2, 88-95.

Gopalan.M.A,Vidhyalakshmi.S,Thiruniraiselvi.N,”On strong Diophantine Quadruples with Property D(1)”, The IJST,July 2014,Vol-2, Issue- 7, 17-22. DOI: https://doi.org/10.14445/22315373/IJMTT-V20P514

Published
2015-04-30
How to Cite
Gopalan, M., Vidhyalakshmi, S., & Thiruniraiselvi, N. (2015). CONSTRUCTION OF IRRATIONAL GAUSSIAN DIOPHANTINE QUADRUPLES. International Journal of Engineering Technologies and Management Research, 1(1), 1-7. https://doi.org/10.29121/ijetmr.v1.i1.2015.20