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LATTICE POINTS ON THE HOMOGENEOUS CUBIC
EQUATION WITH FOUR UNKNOWNS
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Article Type: Research Article
Article Citation: Manju Somanath,
Radhika Das, and Bindu V.A. (2020). LATTICE POINTS ON THE HOMOGENEOUS CUBIC
EQUATION WITH FOUR UNKNOWNS 
. International Journal of Research -GRANTHAALAYAH, 8(8),
135 – 139. https://doi.org/10.29121/granthaalayah.v8.i8.2020.932
Received Date: 30 July 2020
Accepted Date: 26 August 2020
Keywords:
Homogeneous Cubic Equation
Lattice Points
Integral Solutions
Special Polygonal Numbers
Notation
= Polygonal number of rank n
with m sides.
= Three-dimensional figurate
number of rank n with m sides.
= Star number.
= Rhombic Dodecahedral number of rank n.
ABSTRACT
The Homogeneous cubic equation with four unknowns
represented by the equation 
is analyzed for its patterns of non zero
distinct integral solutions. Here we exhibit four different patterns. In each
pattern we can find some interesting relations between the solutions and
special numbers like Polygonal number, Three-Dimensional Figurate number, Star
number, Rhombic Dodecahedral number etc.
1. INTRODUCTION
Number theory, called the Queen of Mathematics, is a broad and diverse part of Mathematics that developed from the study of the integers. Diophantine equation is one of the oldest branches of Mathematics. Diophantine problems dominated most of the unsolved mathematical problems.
The cubic equation offers an unlimited field of research
because of their variety. This paper concerns with an interesting equation
, 
representing a homogeneous cubic equation with
four unknowns for finding its infinitely many solutions and some interesting
relations between the solutions and special numbers like Polygonal number,
Rhombic Dodecahedral number, Star number, Three Dimensional Figurate number.
2. METHOD OF ANALYSIS
Consider the Homogeneous
cubic equation, 
(1)
2.1. PATTERN 1
Introduction of the transformation 
in 
leads to
(3)
Assume 
and write 
(5)
Using 
and 
in 
and using method of factorization, define
(6)
Equating real and imaginary parts on both sides of 
,
we get
Substituting 
and 
in 
,
we obtain the solutions of 
as
Some properties for the above solution are listed below:
is a multiple of 
is a Nasty Number
2.2. PATTERN 2
Introduction of the transformation 
in leads to
(8)
Assume 
and write 
(10)
Using 
and 
in 
and using method of factorization, define
(11)
Equating real and imaginary parts on both sides of 
,
we get
(12)
(13)
Since our aim is to find integer values for the solution,
put 
and 
in
and
,
we get
(14)
(15)
Substituting 
and
in
,
we obtain the solutions of as
Some properties for the above solution are listed below:
.
is a Nasty Number.
2.3. PATTERN 3
Introduction of the transformation 
in leads to
(17)
Assume 
and write 
(19)
Using
and
in
and using method of factorization, define
(20)
Equating real and imaginary parts on both sides of 
,
we get
(21)
(22)
Since our aim is to find integer values for the solutions,
put 
and 
in
and
,
we get
(23)
(24)
Substituting 
and
in
,
we obtain the solutions of as
Some properties for the above solution are listed below:
2.4. PATTERN 4
Introduction of the transformation 
in leads to
(26)
Put 
in 
,
we get 
(28)
Assume 
and write 
(30)
Using 
and 
in 
and using method of factorization, define
(31)
Equating real and imaginary parts on both sides of 
,
we get
(32)
(33)
Since our aim is to find integer values for the solutions,
put 
and 
in
and
,
we get
(34)
(35)
Substituting 
and
in
,
we obtain the solutions of as
Some properties for the above solution are listed below:
, a Nasty Number
, a Perfect Square
3. CONCLUSION
Diophantine Equation are rich in variety. To conclude, one may search for several other patterns of solutions and their properties.
SOURCES OF FUNDING
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
CONFLICT OF INTEREST
The author have declared that no competing interests exist.
ACKNOWLEDGMENT
None.
REFERENCES
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[4]
M. Somanath and J.Kannan,
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© Granthaalayah 2014-2020. All Rights Reserved.