Copyright to IJIRSET www.ijirset.com 1301
OBSERVATIONS ON THE
NON-HOMOGENEOUS SEXTIC EQUATION
WITH FOUR UNKNOWNS
w
z
k
y
x
3
3
2
(
2
3
)
5S.Vidhyalakshmi
1,M.A.Gopalan
2, A.Kavitha
3Professor, PG and Research Department of Mathematics,Shrimati Indira Gandhi college, Trichy-620 002, Tamilnadu,
India1
Professor, PG and Research Department of Mathematics,Shrimati Indira Gandhi college, Trichy-620 002, Tamilnadu,
India2
Lecturer, PG and Research Department of Mathematics,Shrimati Indira Gandhi college, Trichy-620 002, Tamilnadu,
India3
Abstract: The sextic non-homogeneous equation with four unknowns represented by the Diophantine
equation
x
3
y
3
2
(
k
2
3
)
z
5w
is analyzed for its patterns of non-zero distinct integral solutions are illustrated. Various interesting relations between the solutions and special numbers, namely polygonal numbers, Pyramidal numbers, Jacobsthal numbers, Jacobsthal-Lucas number, Pronic numbers, Star numbers are exhibited.Keywords: Integral solutions, sextic non-homogeneous equation.
M.sc 2000 mathematics subject classification: 11D41
NOTATIONS
t
m,n : Polygonal number of rank n with size m
CP
n,m : Centred polygonal number of rank n
Ky
n : Kynea number of rank n
CP
nm : Centred pyramidal number of rank n
F
4,n,6 : Four dimensional figurative number of rank n whose generating polygon ishexagon
F
4,n,7 : Four dimensional figurative number of rank n whose generating polygon isheptagon
S
n : Star number of rank n
Pr
n : Pronic number of rank n
j
n : Jacobsthal lucas number of rank n
J
n : Jacobsthal number of rank n
I. INTRODUCTION
The theory of Diophantine equations offers a rich variety of fascinating problems
[1-4]. Particularly, in [5-6], sextic equations with 3 unknowns are studied for their integral solutions.
[7-9] analyze sextic equations with 4 unknowns for their non-zero integer solutions. This
Copyright to IJIRSET www.ijirset.com 1302 . Various interesting properties among theProperties among the values of x,y,z,w are presented.
II. METHOD OF ANALYSIS
The equation under consideration is
x
y
k
z
w
5 2
3
3
2
(
3
)
(1)
where k is a given non-zero integers
A.Case:1
Introduction of the transformations
,
v
u
x
y
u
v
,
w
2
2s2u
(2)In (1) leads to
2 2 5 2
2
2
3
v
(
k
3
)
z
2
su
(3)Let
z
a
2
3
b
2 (4)Using (4) in (3) and applying the method of factorization, define
)
3
2
2
(
)
3
)(
3
(
3
v
k
i
a
i
b
5 si
si
u
Equating real and imaginary parts, we get
)
9
30
5
)(
2
2
(
3
)
45
30
)(
3
2
2
(
k
a
5a
3b
2ab
4k
a
4b
a
2b
3b
5u
s
s
s
s
(5)
)
9
30
5
)(
3
2
2
(
)
45
30
)(
2
2
(
k
a
5a
3b
2ab
4k
a
4b
a
2b
3b
5v
s
s
s
s
(6)
Using (5) and (6) in (2), we have
)
7
(
)}
9
30
5
)(
1
(
3
)
45
30
)(
3
{(
2
)
,
(
)
9
30
5
(
2
)
45
30
(
2
)
,
(
)
9
30
5
)(
3
(
2
)
45
30
)(
1
(
2
)
,
(
5 3 2 4 4 2 3 5 2 3 5 3 2 4 2 4 2 3 5 2 5 3 2 4 1 4 2 3 5 1
b
b
a
b
a
k
ab
b
a
a
k
b
a
w
b
b
a
b
a
k
ab
b
a
a
b
a
y
b
b
a
b
a
k
ab
b
a
a
k
b
a
x
s s s s sThus (7) and (4) represent the non-trivial integral solutions of (1).
B.Properties:
(
1
)[
90
45
76
]
)
1
,
2
(
)
1
(
x
sk
j
6s 1J
4s 1j
2s 1
(
3
)[
5
(
(
1
)
)
30
(
3
(
1
)
)
9
(
(
1
)
)]
1 1 1 3 1 3 1 5 1
5
ss s
s s
s
J
j
j
k
1
]
2
Pr
18
6
24
24
(
5
)[
3
{(
2
)
,
1
(
)
2
(
w
n
3s2k
F
4,n,7
F
4,n,6
CP
n26
n
t
4,n
3
(
1
)[
(
6
3
)
6
16
Pr
16
4,]
27 10
8 ,
4n
CP
nCP
nCP
n nt
nt
k
Copyright to IJIRSET www.ijirset.com 1303
)
2
(mod
0
)
2
,
2
(
)
3
(
y
n n
For illustration and clear understanding, substituting s=1 in (7), the corresponding non-zero distinct integral solutions to (1) are given by
)
9
30
5
)(
6
2
(
)
45
30
)(
2
2
(
)
,
(
a
b
k
a
5a
3b
2ab
4k
a
4b
a
2b
3b
5x
)
9
30
5
(
4
)
45
30
(
4
)
,
(
a
b
a
5a
3b
2ab
4k
a
4b
a
2b
3b
5y
2 2
3
)
,
(
a
b
a
b
z
)
9
30
5
)(
3
3
(
)
45
30
)(
3
[(
4
)
,
(
a
b
k
a
5a
3b
2ab
4k
a
4b
a
2b
3b
5w
C.Properties:
)]
1
3
(
45
)
1
)
1
(
(
30
1
)[
6
2
(
)
2
,
1
(
)
2
,
1
(
)
1
(
x
n
y
n
k
ky
2n
j
n1
n1
J
4,n
(
6
6
)[
2
(
5
30
(
(
1
)
)
9
(
23
)]
2
2n n n n
n
j
ky
J
k
]
16
135
)
(
30
)[
6
2
(
)
2
,
1
(
)
2
,
1
(
)
2
(
x
n
y
n
k
ky
2n
j
n1
J
4n
(
6
6
)[
2
(
30
2n35
)
9
(
2n3
n)]
n
j
ky
J
k
]
24
48
3
)
)(
)[(
3
(
4
)
1
,
(
)
3
(
w
n
k
t
4,nCP
n6
CP
n28
CP
n4
t
3,n
t
4,n
4
(
3
3
)[
24
6
2
8
3,33
4,9
]
6 11
7 , ,
4
k
F
nCP
nCP
nt
nt
n]
1
5
15
)
(
5
)[
3
[(
6
)
,
1
(
)
,
1
(
)
,
1
(
)
4
(
x
n
y
n
w
n
k
t
4,nt
20,n
CP
n16
t
14,n
(
3
3
)[
(
3
2
)
6
26
3,13
4,]
24 9
16 ,
4n
CP
nCP
nCP
nt
nt
nt
k
D.Case:2
Consider the transformations
v
u
x
,y
u
v
,w
u
(8) Using (8) in (1), we have
5 2
2
2
3
v
(
k
3
)
z
u
(9)Employing the factorization method, define
5
)
3
)(
3
(
3
v
k
i
a
i
b
i
u
Equating real and imaginary parts, one has
)
9
30
5
(
3
)
45
30
(
a
5a
3b
2ab
4a
4b
a
2b
3b
5k
u
)
9
30
5
(
)
45
30
(
a
5a
3b
2ab
4k
a
4b
a
2b
3b
5v
Thus, the non-zero distinct integral solutions to (1) are given by
)
9
30
5
)(
3
(
)
45
30
)(
1
(
)
,
(
a
b
k
a
5a
3b
2ab
4k
a
4b
a
2b
3b
5x
)
9
30
5
)(
3
(
)
45
30
)(
1
(
)
,
(
a
b
k
a
5a
3b
2ab
4k
a
4b
a
2b
3b
5y
2 2
3
)
,
(
a
b
a
b
z
)
9
30
5
(
3
)
45
30
(
)
,
(
a
b
k
a
5a
3b
2ab
4a
4b
a
2b
3b
5w
Copyright to IJIRSET www.ijirset.com 1304
]
22
22
2
6
)
2
(
)[
1
(
)
1
,
(
)
1
(
x
n
k
t
4,nCP
n3
CP
n28
CP
n9
t
3,n
t
4,n
(
3
)[
(
2
)
6
2
3,31
4,9
]
8 ,
12 ,
4
k
t
nt
nCP
nt
nt
n]
22
44
2
6
)
)(
2
[(
2
)
1
,
(
)
1
,
(
)
2
(
x
n
y
n
t
4,nCP
n3
CP
n28
CP
n9
t
3,n
t
4,n
2
[(
10
)(
)
5
30
4,9
]
6 ,
3 ,
4n
t
n
CP
n
t
n
t
k
]
1
30
)
5
38
2
(
[
)
,
1
(
)
,
1
(
)
3
(
y
n
w
n
t
4,nt
22,n
t
3,n
t
4,n
t
4,n
[
(
6
)
26
10
3,5
4,]
6 12
11 ,
4n
CP
nCP
nCP
nt
nt
nt
k
]
23
46
6
)
2
(
[
3
)
1
,
(
)
1
,
(
)
1
,
(
)
4
(
x
n
y
n
w
n
k
t
4,nCP
n12
CP
n3
CP
n28
t
3,n
t
4,n
9
[(
)(
)
6
2
3,31
4,9
]
8 ,
12 ,
4
t
nt
nCP
nt
nt
n]
1
6
12
2
2
6
6
)
(
)[
1
(
)
,
1
(
)
5
(
x
n
k
t
4,nS
n
CP
n28
CP
n17
t
23,n
t
12,n
t
3,n
t
4,n
(
3
)[(
6
)(
)
27
10
3,5
4,]
6 9
,
4n
CP
nCP
nt
nt
nt
k
F.Case:3
Write (9) as
1
*
)
3
(
3
2 2 52
v
k
z
u
(10)Write 1 as
4
)
3
1
)(
3
1
(
1
i
i
(11)
Substituting (4) and (11) in (10) and employing the factorization method, define
5
)
3
)(
3
(
2
)
3
1
(
3
v
i
k
i
a
i
b
i
u
Equating real and imaginary parts, we get
)]
9
30
5
)(
1
(
3
)
45
30
)(
3
[(
2
1
5 3 2 4 4 2 3 5b
b
a
b
a
k
ab
b
a
a
k
u
)]
9
30
5
)(
3
(
)
45
30
)(
1
[(
2
1
5 3 2 4 4 2 3 5b
b
a
b
a
k
ab
b
a
a
k
v
Thus, taking a=2A, b=2B the non zero distinct integral solutions to (1) are given by
)]
9
30
5
)(
3
(
)
45
30
)(
1
[(
2
)
,
(
A
B
5k
A
5A
3B
2AB
4K
A
4B
A
2B
3B
5x
)]
9
30
5
(
)
45
30
[(
2
)
,
(
A
B
6A
5A
3B
2AB
4K
A
4B
A
2B
3B
5y
)
3
(
2
)
,
(
A
B
2A
2B
2z
)]
9
30
5
)(
1
(
3
)
45
30
)(
3
[(
2
)
,
(
A
B
4k
A
5A
3B
2AB
4K
A
4B
A
2B
3B
5Copyright to IJIRSET www.ijirset.com 1305 It is to be noted that one may get integral solutions to (1) when k takes odd values.
G.Properties:
2
{(
1
)[
45
(
6
2
3
)
30
1
]
)
,
1
(
)
1
(
x
n
5k
F
4,n,6CP
n9t
4,nt
4,n
(
3
)[
(
3
2
52
3,26
4,)
10
3,5
4,]}
3 16
,
4n
CP
nCP
nt
nt
nt
nt
nt
k
]
22
44
2
6
)
2
(
{[
2
)
1
,
(
)
2
(
y
n
6t
4,nCP
n3
CP
n28
CP
n9
t
3,n
t
4,n
[
30
10
10
3,35
4,9
]}
9 6
, ,
4
k
F
nCP
nt
nt
n
2
{(
3
)(
15
(
)
30
30
1
)
)
,
1
(
)
3
(
w
n
4k
t
4,nt
8,nCP
n6t
4,n
3
(
1
)[
6
(
)
6
32
3,16
4,]}
27 9
,
4n
CP
nCP
nt
nt
nt
k
]
1
57
)
8
2
6
(
9
)[
9
3
{(
2
)
,
1
(
)
,
1
(
)
,
1
(
)
4
(
x
n
y
n
w
n
4k
F
4,n,7
Cp
n13
CP
n3
t
3,n
t
4,n
]}
12
24
6
)
)(
(
)
)(
)[(
9
9
(
k
CP
n6t
20,n
t
4,nt
18,n
CP
n23
t
3,n
t
4,n
H.Case:4
Instead of (11) we write 1 as
49
)
3
4
1
)(
3
4
1
(
1
i
i
Following the procedure as presented in pattern 4 the corresponding non- zero distinct integral solutions to (1) are obtained as
)]
9
30
5
)(
15
11
(
)
45
30
)(
11
5
[(
7
)
,
(
A
B
4k
A
5A
2B
2AB
4k
A
4B
A
2B
3B
5x
)]
9
30
5
)(
9
13
(
)
45
30
)(
13
3
[(
7
)
,
(
A
B
4k
A
5A
2B
2AB
4k
A
4B
A
2B
3B
5y
)
3
(
7
)
,
(
A
B
2A
2B
2z
)]
9
30
5
)(
1
4
(
3
)
45
30
)(
12
[(
7
)
,
(
A
B
4k
A
5A
2B
2AB
4k
A
4B
A
2B
3B
5w
I.Properties:
7
[(
5
11
)(
(
6
)
6
6
44
22
)
)
1
,
(
)
1
(
x
n
4k
t
4,nCP
n3CP
n28CP
n8t
3,nt
4,n
(
11
15
)[
5
(
12
6
24,18
3,9
4,)
14
]
8 4
, ,
4
F
nCP
nCP
nt
nt
nk
7
{(
3
13
)[
15
(
)
15
(
3
4
)
1
]
)
,
1
(
)
2
(
y
n
4k
t
4,nCP
6,nCP
n10t
3,nt
4,n
(
13
9
)[
6
(
)
6
32
3,16
4,]}
2 9
,
4n
CP
nCP
nt
nt
nt
k
)
2
,
2
(
)
3
(
z
n n is a perfect square.]
15
16
2
6
24
[
9
){
12
[(
7
)
,
1
(
)
4
(
w
n
4k
F
4,n,7
CP
n13
CP
n3
t
3,n
t
4,n]]
5
10
)
10
2
4
(
3
)[
4
1
(
3
}
6
]
6
12
[
5
S
n
t
3,n
t
4,n
k
CP
n6t
8,n
t
3,n
t
4,n
t
3,n
t
4,nCopyright to IJIRSET www.ijirset.com 1306
J.Case:5
Consider the transformations
v
u
x
,y
u
v
,w
(
3
s
2
6
s
4
)
u
(12)Repeating the process as in pattern 1, the non-zero distinct integral solutions to (1) are obtained as
(
3
1
)(
30
45
)
)
,
(
a
b
k
ks
s
a
5a
3b
2ab
4x
(
k
3
s
3
3
ks
)(
5
a
4b
30
a
2b
3
9
b
5)
(
3
1
)(
30
45
)
)
,
(
a
b
k
ks
s
a
5a
3b
2ab
4y
(
3
3
3
)(
5
30
9
)
5 3 2
4
b
a
b
b
a
ks
s
k
2 2
3
)
,
(
a
b
a
b
z
(
3
6
4
){(
3
)(
30
45
)
)
,
(
a
b
s
2s
k
s
a
5a
3b
2ab
4w
3
(
1
)(
5
30
9
)
5 3 2
4
b
a
b
b
a
ks
Where s =2,3,4…………
K.Properties:
]
1
15
)
18
2
6
24
24
(
5
)[
1
3
(
)
,
1
(
)
1
(
x
n
k
ks
s
F
4,n,8
F
4,n,5
CP
n26
t
17,n
t
3,n
t
4,n
(
3
3
3
)[
3
(
)[
2
18
3,9
4,]
10
3,5
4,]
9 ,
4n
CP
nt
nt
nt
nt
nt
ks
s
k
]
1
)
4
3
(
15
)
(
15
)[
1
3
(
)
,
1
(
)
2
(
y
n
k
s
ks
t
4,nCP
6,n
CP
n10
t
3,n
t
4,n
(
3
3
3
)[(
3
)(
8,4
3,2
4,10
)
10
3,5
4,]
6
n n
n n
n
n
t
t
t
t
t
CP
s
k
ks
(
3
6
4
){(
3
)[
(
2
)
6
2
22
22
]
)
1
,
(
)
3
(
w
n
s
2s
k
s
t
4,nCP
n3CP
n28CP
n9t
3,nt
4,n
3
(
1
)[
30
10
10
3,35
4,9
]}
9 6
, ,
4
ks
F
nCP
nt
nt
n
III. CONCLUSION
To conclude one may search for other pattern of solutions and their corresponding properties.
REFERENCES
[1]. L.E.Dickson, History of Theory of Numbers, Vol.11, Chelsea Publishing company, New York (1952).
[2]. L.J.Mordell, Diophantine eequations, Academic Press, London (1969).
[3]. Telang,S.G., Number theory, Tata Mc Graw Hill publishing company, New Delhi (1996)
[4]. Carmichael, R.D., The theory of numbers and Diophantine Analysis, Dover Publications, New York (1959).
[5]. M.A.Gopalan and Sangeetha.G., On the sextic equations with three unknowns
x
2
xy
y
2
(
k
2
3
)
nz
6 , Impact J.Sci.tech. Vol.4, No:4, 89-93(2010).Copyright to IJIRSET www.ijirset.com 1307 [7]. M.A.Gopalan and A.VijayaSankar, Integral Solutions of the sextic equation
6 4
4
4
y
z
2
w
x
,Indian Journal of Mathematics and mathematical sciences, Vol.6, No2, 241-245,(2010). [8]. M.A.Gopalan, S.Vidhyalakshmi and A.Vijayasankar, Integral solutions of hon- homogeneous
sextic equation
