Construction of Gaussian Diophantine triples with the property D (25)

We search for Gaussian Diophantine triples with the property D(25) and we present 11 general sets of Gaussian Diophantine triples with the property D(25).

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Introduction

The Gaussian integer z[i] is the simplest generalization of ordinary integer Z and they behave in much the same way. This was first introduced by Carl Friedrich Gaussian in his second monograph on quartic reciprocity in the year 1832. In particular z[i] enjoys unique prime factorization and this allows us reason z[i] as the same way we do about Z. For basics and ideas [1-3] has been studied.

For various ideas on triples and quadruples [4-13] has been referred. [14, 15] has been studied for ideas on Gaussian Diophantine quadruples. CC Recently in [16, 17] Special dio 3-tuples are constructed from n Gno n In this paper search for Gaussian Diophantine triples with the property D(25) and we present 11 general sets of Gaussian Diophantine triples with the property D(25). Definition Gaussian integers are defined as z[i] = } , ; { Z b a ib a   A sequence of non-negative integers is said to be Diophantine m-triple if each ) ( 1 j i a a j i   is the square of an integer. A Diophantine 3-triple is also known as Diophantine triples.

Method of Analysis Let ikq kp a    1 ; ikq kp b    11 be two Gaussian integers. We have 2 25    ab (1) where ikq kp    6  Hence (a,b) is a Gaussian Diophantine double with the property D(25) Introduce c such that c is any non-zero Gaussian Integer with the property. 2 25    ac (2) 2 25    bc (3) Taking         b and a , we have from (2) and (3) kq i kp c 4 24 4    Thus we have the Gaussian Diophantine triples } 4 24 4, 11 , 1 { kq i kp ikq kp ikq kp       with the property D(25) for any k > 0. Maths 2017; 2(6):301-302 P Saranya Assistant Professor, Department of Mathematics, Cauvery College for Women, Trichy, Tamil Nadu, India G Janaki Associate Professor, Department of Mathematics, Cauvery College for Women, Trichy, Tamil Nadu, India Correspondence P Saranya Assistant Professor, Department of Mathematics, Cauvery College for Women, Trichy, Tamil Nadu, India D(4),International Journal of Physics and Mathematical Sciences. 2014; 4(2):65-67. 16. Janaki G, Saranya P.

Construction of Special Dio 3- In the following table we present 10 sets of Gaussian Diophantine triples with the property D (25) Tuples from CC - I”, International Journal of Advanced n Gno n S. No a B C 1 ikq kp  1 ikq kp  9 kq i kp 4 16 4   2 ikq kp  2 ikq kp  8 kq i kp 4 12 4   Research and Development, 2017, l-26, 151-154. 17. Janaki G, Saranya P. Construction of Special Dio 3- Tuples from CC - II, International Journal for Research 3 ikq kp  3 ikq kp  7 kq i kp 4 8 4   n Gno n 4 ikq kp  4 ikq kp  6 kq i kp 4 4 4   5 ikq kp  6 ikq kp  4 kq i kp 4 4 4   in Applied Science and Engineering Technology. 2017; 5(11):1642-1645. 6 ikq kp  7 ikq kp  3 kq i kp 4 8 4   7 ikq kp  8 ikq kp  2 kq i kp 4 24 4   8 ikq kp  9 ikq kp  1 kq i kp 4 16 4   9 ikq kp  12 ikq kp  2 kq i kp 4 28 4   10 ikq kp  2 ikq kp  12 kq i kp 4 28 4  

Conclusion

In this paper we have presented 11 sets of Gaussian triples with the property D(25). One may also search for similar Gaussian triples with suitable property.

Experimental

Math. 2008; 17(1). 6. Gopalan MA, Srividhya G. Diophantine Quadruple for Fibonacci and Lucas Numbers with property D(4), Diophantus Journal of Mathematics. 2012; 1(1):15-18, 7. Gopalan MA, Srividhya G.

Some non-extendable P-5 sets, Diophantus Journal of Mathematics. 2012; 1(1):19-22. 8. Gopalan MA, Srividhya G. Two Special Diophantine Triples, Diophantus Journal of Mathematics. 2012; 1(1):23-27. 9. Gupta H, Singh K.

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Tech. 2012; 33:253-256. 12. Deshpande MN. Families of Diophantine Triplets, Bulletin of the Marathwada Mathematical Society.2003; 4:19-21. 13. Fujita Y.

The extensibility of Diophantine pairs (k- 1,k+1), Journal of Number Theory. 2008; 128:322- 353, 14. Vidhyalakshmi S, Gopalan MA, Lakshmi K. Gaussian Diophantine quadruples with property D(1), IOSR Journal of Mathematics. 2014; 10(3):II, 12-14 15. Gopalan MA, Vidhyalakshmi S, Premalatha E, Presenna R.

Gaussian Diophantine quadruples with property

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