The document analyzes solutions to the homogeneous cubic equation with four unknowns    2 2 3 3 16zw y x y x y x     . Three methods are presented to obtain distinct integer solutions: 1) Using factorization and real/imaginary parts, 2) Introducing linear transformations, 3) Writing the equation in terms of ratios. Each method yields multiple sets of integer solutions and some solutions are related to special number patterns like polygonal, gnomonic, star, and pronic numbers.

1. *Corresponding Author: Sharadha Kumar, Email: sharadhak12@gmail.com
RESEARCH ARTICLE
Available Online at www.ajms.in
Asian Journal of Mathematical Sciences 2018; 2(1):19-23
On Homogeneous Cubic Equation with Four Unknowns
   2
2
3
3
16zw
y
x
y
x
y
x 




S.Vidhyalakshmi1
, M.A.Gopalan2
, Sharadha Kumar3*
1,2
Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu,
India.
3*
M.Phil scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu,
India
Received on: 11/12/2017,Revised on: 30/12/2017,Accepted on: 15/01/2018
ABSTRACT
The cubic diophantine equation with four unknowns given by    2
2
3
3
16zw
y
x
y
x
y
x 



 is
analyzed for its non-zero distinct integer points on it. Different patterns of integer points for the equation
under consideration are obtained. A few interesting relations between the solutions and special number
patterns namely Polygonal number, Gnomonic number, Star number and Pronic number are presented.
Keywords: cubic equation with four unknowns, integral solutions
2010 Mathematics Subject Classification: 11D25
INTRODUCTION
The cubic diophantine equations offer an unlimited field for research due to their variety [1, 22]
. For an
extensive review of various problems, one may refer [2-21]
. This communication concerns with yet another
interesting cubic diophantine equation with four unknowns    2
2
3
3
16zw
y
x
y
x
y
x 



 for
determining its infinitely many non-zero integral points. Also, a few interesting relations between the
solutions and special numbers are presented.
Notations:
 Polygonal number of rank n with size m
  





 



2
2
1
1
,
m
n
n
t n
m
 Gnomonic number of rank n
1
2 
 n
GNOn
 Star number of rank n
1
6
6 2


 n
n
Sn
 Pronic number of rank n
 
1

 n
n
PRn
Method of analysis:
The homogeneous cubic equation with four unknowns to be solved is
   2
2
3
3
16zw
y
x
y
x
y
x 



 (1)
Introducing the linear transformations
u
z
v
u
y
v
u
x 



 ,
, (2)
in (1), it gives
2
2
2
8
7 w
v
u 
 (3)
Assume

2. Kumar Sharadha et al./ On Homogeneous Cubic Equation with Four Unknowns    2
2
3
3
16zw
y
x
y
x
y
x 




© 2017, AJMS. All Rights Reserved 20
0
,
,
7 2
2


 b
a
b
a
w (4)
We present below different methods of solving (3) and thus, different sets of non-zero distinct integer
solutions to (1) are obtained.
Method: 1
Write 8 as
  
7
1
7
1
8 i
i 

 (5)
Substituting (4), (5) in (3) and employing the method of factorization, define
    2
7
7
1
7 b
i
a
i
v
i
u 


 (6)
Equating the real and imaginary parts in (6), it is seen that
ab
b
a
v
ab
b
a
u
2
7
14
7
2
2
2
2






In view of (2), we have
 
 
  

















ab
b
a
b
a
z
z
ab
b
a
y
y
ab
b
a
b
a
x
x
14
7
,
16
,
12
14
2
,
2
2
2
2
(7)
Thus, (4) and (7) represent the integer solutions to (1)
Properties:

  0
17
2
30
1
, 




 a
a
a GNO
PR
S
a
a
x

   
2
mod
0
31
8
,
1 ,
66 



 b
PR
t
b
b
x b
b

   
3
mod
0
11
20
,
1 


 b
b GNO
PR
b
b
z

     
7
mod
0
1
,
1
, 


 a
PR
a
z
a
x a
Method: 2
Introducing the linear transformations



 8
,
7 


 v
w
in (3), it is written as
2
2
2
56 u

 
 (8)
which is satisfied by
2
2
2
2
56
,
56
,
2 s
r
s
r
u
rs 



 

The corresponding integer solutions to (1) are found to be
 
 
 
  rs
s
r
s
r
w
w
s
r
s
r
z
z
rs
s
s
r
y
y
rs
r
s
r
x
x
14
56
,
56
,
16
2
,
16
112
,
2
2
2
2
2
2














Properties:

  r
r PR
GNO
r
r
x 128
64
64
, 



   
7
mod
0
2
,
1 
 s
PR
s
y

  0
55
55
, 

 s
PR
s
s
z s

   
2
mod
0
5
24
1
, 



 s
s
s GNO
PR
S
s
s
y
It is to be noted that (8) is written as the system of double equations as follows:
System 1 2 3 4 5 6 7
u

 2
 2
2 2
4 
56 
28 
14 
8
u

 56 28 14  
2 
4 
7
AJMS,
Jan-Feb,
2018,
Vol.
2,
Issue
1

3. Kumar Sharadha et al./ On Homogeneous Cubic Equation with Four Unknowns    2
2
3
3
16zw
y
x
y
x
y
x 




© 2017, AJMS. All Rights Reserved 21
Solving each of the above systems, and performing a few calculations, the corresponding solutions to (1)
are represented as shown below:
Solution to system 1:
k
k
w
k
z
k
y
k
k
x 14
28
2
,
28
2
,
16
56
,
16
4 2
2
2










Solution to system 2:





 7
14
,
14
,
8
28
,
8
2 2
2
2









 w
z
y
x
Solution to system 3:





 7
7
2
,
7
2
,
8
14
,
8
4 2
2
2









 w
z
y
x
Solution to system 4:
k
w
k
z
k
y
k
x 71
,
55
,
18
,
128 




Solution to system 5:



 22
,
13
,
10
,
36 



 w
z
y
x
Solution to system 6:



 16
,
5
,
12
,
22 



 w
z
y
x
Solution to system 7:
k
w
k
z
k
y
k
x 29
,
,
30
,
32 




Method: 3
Observe that (3) is written as
 
2
2
2
2
7 v
w
w
u 

 (9)
Write (9) in terms of ratio as
  0
,
7










w
u
v
w
v
w
w
u
which is equivalent to the system of double equations
 
  0
7
7
0








w
v
u
w
v
u








Applying the method of cross multiplication, we have



















2
7
14
7
2
2
2
2
v
u
(10)
2
2
7
 


w (11)
In view of (2), we have
























14
7
16
12
14
2
2
2
2
2
z
y
x
(12)
Thus (11) and (12) represent the integer solutions of (1)
Note: 1
Also, (9) is written in the form of ratios as shown below:
i.  
0
,
7










w
u
v
w
v
w
w
u
ii.  
0
,
7










w
u
v
w
v
w
w
u
iii.
  0
,
7










w
u
v
w
v
w
w
u
Following the analysis as in Method 3, one gets three more sets of integer solutions to (1).
Properties:

   
3
mod
0
13
14
,
1 

 

 GNO
PR
x

   
2
mod
0
16
, 
 

 PR
y
AJMS,
Jan-Feb,
2018,
Vol.
2,
Issue
1

4. Kumar Sharadha et al./ On Homogeneous Cubic Equation with Four Unknowns    2
2
3
3
16zw
y
x
y
x
y
x 




© 2017, AJMS. All Rights Reserved 22

   
11
mod
0
4
8
1
, 


 


 GNO
PR
z

   
3
mod
0
5
2
1
, 

 

 GNO
PR
x
Remarkable Observations:
Let  
0
0
0 ,
, w
v
u be the initial solution of (3). Let  
1
1
1 ,
, w
v
u be the second solution of (3), where
0
1
0
1
0
1 ,
, w
h
w
h
v
v
u
u 



 (13)
0
0
1
0
0
1
15
14
16
15
w
v
w
w
v
v





The above two equations is written in the matrix form as

























0
0
1
1
15
16
14
15
w
v
w
v
where 








15
16
14
15
M
Repeating the above process, the general values of vand ware given by

















0
0
w
v
M
w
v n
n
n
If 
, are the eigen values of M , then 14
4
15
,
14
4
15 


 

 
 
 
 
I
M
I
M
M
n
n
n















 
 


















2
14
2
8
14
2
n
n
n
n
n
n
n
n
n
M








The general values of z
y
x ,
, satisfying (1) are
  n
n
n
n
n
n
n w
v
u
x
14
2
2
0



 




  n
n
n
n
n
n
n w
v
u
y
14
2
2
0



 




0
u
zn 
  n
n
n
n
n
n
n w
v
w
2
8
14 


 



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5. Kumar Sharadha et al./ On Homogeneous Cubic Equation with Four Unknowns    2
2
3
3
16zw
y
x
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x
y
x 


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© 2017, AJMS. All Rights Reserved 23
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