77 sciencegiven point on the considered cone are exhibited. 2. method of analysis the ternary...
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RESEARCH
Gopalan et al.Lattice points on the homogeneous cone, z2 = 4x2 + 10y2,Indian Journal of Science, 2012, 1(2), 89-91, www.discovery.org.inhttp://www.discovery.org.in/ijs.htm © 2012 discovery publication. All rights reserved
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Gopalan MA1, Geetha V2
1. Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichirappalli, Tamilnadu, India, E-mail:[email protected]. 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
ABSTRACTWe obtain infinitely many non-zero distinct integral points on the homogeneous cone given by z2 = 4x2 + 10y2, a few interesting relations between thesolutions and special number patterns are presented.
Keywords: Ternary Quadratic, Lattice Points, Homogeneous Cone.
MSc 2000 Mathematics Subject classification: 11D09
NOTATIONS:Special numbers Notations DefinitionsStar 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. INTRODUCTIONThe ternary quadratic diophantine equations (homogeneous and non-homogeneous) offer an unlimited field for research by reason of variety [1-2]. Foran extensive review of various problems one may refer [3-17]. This communication concerns with yet another interesting ternary quadratic equationrepresenting a homogeneous cone z2 = 4x2 + 10y2 for determining its infinitely many non-zero integral solutions. Also a few interesting relations betweenthe solutions and special number patterns are presented. Further three different general forms for generating sequence of integral points based on thegiven point on the considered cone are exhibited.
2. METHOD OF ANALYSISThe 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 - IIntroducing the linear transformations,
in (1), it is written as
RESEARCH Indian Journal of Science, Volume 1, Number 2, December 2012
Lattice points on the homogeneous cone, z2 = 4x2 + 10y2
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RESEARCH
Gopalan et al.Lattice points on the homogeneous cone, z2 = 4x2 + 10y2,Indian Journal of Science, 2012, 1(2), 89-91, www.discovery.org.inhttp://www.discovery.org.in/ijs.htm © 2012 discovery publication. All rights reserved
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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 - IIEquation (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
RESEARCH
Gopalan et al.Lattice points on the homogeneous cone, z2 = 4x2 + 10y2,Indian Journal of Science, 2012, 1(2), 89-91, www.discovery.org.inhttp://www.discovery.org.in/ijs.htm © 2012 discovery publication. All rights reserved
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4. CONCLUSIONTo conclude one may search for other patterns of solutions and their corresponding properties.
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