Lattice Points on the Homogeneous Cone, Z2 = 4X2 + 10Y2 Science

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RESEARCH Indian Journal of Science, Volume 1, Number 2, December 2012 RESEARCH 49 77 – Indian Journal of EISSN 2319 30 77 – Science ISSN 2319 Lattice points on the homogeneous cone, z2 = 4x2 + 10y2

Gopalan MA1, Geetha V2 1. Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichirappalli, Tamilnadu, India, E-mail:[email protected] 2. Asst.Professor, Department of Mathematics, Cauvery College for Women,Trichirappalli,Tamilnadu,India,E-mail:[email protected]

Received 08 October; accepted 12 November; published online 01 December; printed 16 December 2012 ABSTRACT We obtain infinitely many non-zero distinct integral points on the homogeneous cone given by z2 = 4x2 + 10y2, a few interesting relations between the solutions and special number patterns are presented. Keywords: Ternary Quadratic, Lattice Points, Homogeneous Cone. MSc 2000 Mathematics Subject classification: 11D09 NOTATIONS: Special numbers Notations Definitions Star number Sn

Gnomonic number Gn

Regular Polygonal number tm,n

Pronic number Pn

Decagonal number Dn

Tetra decagonal number TDn

Tetrahedral number THn

Truncated Tetrahedral number TTn

Truncated Octahedral number TOn Centered Cube number CCn

Rhombic dodagonal number Rn

Triangular number Tn

Woodall number Wn

Centered Hex number Ct6,n 1. INTRODUCTION The ternary quadratic diophantine equations (homogeneous and non-homogeneous) offer an unlimited field for research by reason of variety [1-2]. For an extensive review of various problems one may refer [3-17]. This communication concerns with yet another interesting ternary quadratic equation representing a homogeneous cone z2 = 4x2 + 10y2 for determining its infinitely many non-zero integral solutions. Also a few interesting relations between the solutions and special number patterns are presented. Further three different general forms for generating sequence of integral points based on the given point on the considered cone are exhibited. 2. METHOD OF ANALYSIS The Ternary quadratic equation representing homogeneous equation is

To start with, it is seen that (1) is satisfied by the following triples: (3, 4, 14) and (5r2-2s2, 4rs, 10r2+4s2) However, we have other patterns of solutions which are illustrated as follows. 2.1. Pattern - I Introducing the linear transformations,

in (1), it is written as 89

Gopalan et al. Lattice points on the homogeneous cone, z2 = 4x2 + 10y2, Indian Journal of Science, 2012, 1(2), 89-91, www.discovery.org.in http://www.discovery.org.in/ijs.htm © 2012 discovery publication. All rights reserved RESEARCH

Write 14 as

Using (4)and (5) in (3) and employing the method of factorization, define

Equating real and imaginary parts, we get

2.2. Properties

It is to be noted that in (2) we may also take,

For this Choice, the corresponding integral points on (1) are obtained as

2.3. Pattern - II Equation (3) is written as

Write 40 as

Using (8) and (9) in (7) and employing the method of factorization, define

Equating rational and irrational parts, we get

2.4. Properties

3. GENERATION OF SOLUTIONS 90

4. CONCLUSION To conclude one may search for other patterns of solutions and their corresponding properties. REFERENCES

,1 983.

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