The cubic diophantine equation with four unknowns given by     3 3 2 2 x  y  x  y x  y 16zw 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.

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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