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1. Research Inventy: International Journal Of Engineering And Science
Vol.3, Issue 5 (July 2013), Pp 54-57
Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com
54
On The Non Homogeneous Ternary Quintic Equation
522
7 zyxyx 
M.A.Gopalan, S. Vidhyalakshmi, A.Kavitha, M.Manjula
Department of Mathematics,Shrimati Indira Gandhi College,Tiruchirappalli – 620 002.
ABSTRACT: The ternary Quintic Diophantine Equation given by is
522
z7yxyx  analyzed for its
patterns of non – zero distinct integral solutions. A few interesting relations between the solutions and special
polygonal numbers are exhibited.
KEY WORDS: Quintic equation with three unknowns, Integral solutions.
I. INTRODUCTION
The theory of diophantine equations offers a rich variety of fascinating problems. In particular,quintic
equations, homogeneous and non-homogeneous have aroused the interest of numerous mathematicians since
ambiguity[1-3].For illustration, one may refer [4-9] for quintic equation with three unknowns . This paper
concerns with the problem of determining non-trivial integral solutions of the non-homogeneous quintic
equation with three unknowns given by
522
z7yxyx  .A few relations among the solutions are
presented.
II. NOTATIONS USED
 nm
t , - Polygonal number of rank n with sizem .

m
n
P - Pyramidal number of rank n with sizem .
 nm
ct , - Centered polygonal number of rank n with sizem .
 a
gn - Gnomonic number of rank a
 nso - Stella octangular number of rank n
 n
s - Star number of rank n
 npr - Pronic number of rank n
 npt - Pentatope number of rank n
 nm
CP ,
- Centered pyramidal number of rank n with sizem .
 n
sm
f ,
- m-dimensional figurate number of rank n with s sides.
III. METHOD OF ANALYSIS
The quintic Diophantine equation with three unknowns to be solved for
its non-zero distinct integral solution is
522
z7yxyx  (1)
Introduction of the linear transformation
vux  and vuy  (2)
in (1) leads to
522
z7v3u  (3)
Different patterns of solutions of (3) and hence that of (1) using (2) are given below
1. Pattern -1
Let
22
b3a)b,a(zz  (4)
Write )3i2)(3i2(7  (5)
Using (4) & (5) in (3) and applying the method of factorization, define

2. On The Non homogeneous Ternary Quintic Equation
55
5
)b3ia)(3i2(v3iu 
Equating the real and imaginary parts, we have
54322345
54322345
b18ab45ba60ba30ba10a)b,a(vv
b27ab90ba90ba60ba15a2)b,a(uu


Substituting the above values of u and v in (2), we get
54322345
b9ab135ba30ba90ba5a3)b,a(xx  (6)
54322345
b45ab45ba150ba30ba25a)b,a(yy  (7)
Thus (4), (6) and (7) represent the non-zero distinct solutions of (1) in two parameters.
A few interesting properties observed are as follows:
1. aaa
aa
ttCPffayax ,4,3,76,46,5
533441513012054)1,()1,( 
2. aaaa
aa
ttSSOffazay ,4,36,43,5
2
332668482530120224)1,()1,( 
3. aa
a
tOHfayax ,44,4
545150300126)1,(3)1,( 
4. 1161039)1,(3)1,( 22
,223,4

aaa
tStayax
5. 206gn43t35)1,a(z41)1,a(y3)1,a(x 22
)a2(a,6

6. )2(mod0)1,a(y)1,a(x 
7. )70(mod56)1,(3)1,(  ayax
8. )(mod0),1(),1(3 bbxby 
9. }CP420)1,a(y)1,a(x5{38416 a,6
 is a quintic integer.
2. Pattern-2:
In addition to (5),
Write 7 as )33i1)(33i1(
4
1
7 
Following the procedure similar to Pattern-1, the non-zero distinct integer values of
x , y and z satisfying (1) are given by
22
54322345
54322345
b3a)b,a(zz
b45ab45ba150ba30ba25a)b,a(yy
b36ab90ba120ba60ba20a2)b,a(xx



Properties:
1. aaa
aa
tSOffayax ,46,45,5
118Pr7845901209)1,()1,( 
2. a
a fPatayax
6,4
5
,4 150150395126)1,(2)1,( 
3. 2
a
4
a,3
)(Pr420126)t(140)1),1a(a(y2)1),1a(a(x 
4. bbb SOgnCPbybx 70)1(6335),1(2),1( 632,6 
5. )3(mod0)1,a(z)1,a(y)1,a(x
2

3. Pattern-3:
Also, Write 7 as )3i5)(3i5(
4
1
7 
For this choice of 7, the non-zero distinct integer value of x , y and z satisfying (1) are given by

3. On The Non homogeneous Ternary Quintic Equation
56
22
54322345
54322345
b3a)b,a(zz
b36ab90ba120ba60ba20a2)b,a(yy
b9ab135ba30ba90ba5a3)b,a(xx



Properties:
1. aa
aa
ttPaffayax ,4,3
5
6,47,5
191901041862445)1,()1,( 
2. aaaaa
aa
ttttPaCPffax ,4,9,4,92
5
,36,45,5
721362490729)1,( 2 
3. aaa
aa
tCPffayax ,4,34,47,5
55Pr3405054012027)1,()1,( 
4. )2(mod0)1,a(y3)1,a(x2 
5. )30(mod14)1,a(y)1,a(x4 
6. ]}1[6370),1(3),1(2{8 5 
bb
gnSObybx is a cubical integer
7. Each of the following represents a nasty number:
 }414),1(),1(4{2 24
,422 bb
tgnbybx 
 )a,a(z6
IV. REMARKABLE OBSERVATIONS
I: Consider x and y to be the length and breadth of a rectangle R, whose area, perimeter and length of its
diagonal are respectively denoted by A,P and L.
Then, it is noted that,
 )28(mod012
2
 AP
 )7(mod0AL
2

 }AL{2401
2
 is a quintic integer
II: Employing the integral solutions of (1), a few interesting results among the special numbers are
exhibited.
1)














































 y,3
5
y
2x2,3
4
1x
2
2y
3
2y
2
1x,3
3
x
t
P4
t
P
Pr
P6
t
P3
2401 is a quintic integer.
2) Each of the following represents a nasty number:

5
z
1z2,3
1x,6
4
x
y,3
5
y
2
1x
3
2x
gn
t
42
t
P
t
P4
6
1S
P36
6





































5
2z2,3
4
1z
1y
5
y
x,3
5
x
2y,3
3
1y
4
1y
t
P6
42
1S
P12
t
P
6
t
)PP(3
6

































 





CONCLUSION
To conclude, one may search for other patterns 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 equations, Academic Press, London(1969).
[3]. Carmichael ,R.D.,The theory of numbers and Diophantine Analysis,Dover Publications, New York (1959)
[4]. M.A.Gopalan & A.Vijayashankar, An Interesting Diophantine problem
533
2 zyx 
Advances in Mathematics, Scientific
Developments and Engineering Application, Narosa Publishing House,2010, Pp 1-6.

4. On The Non homogeneous Ternary Quintic Equation
57
[5]. M.A.Gopalan & A.Vijayashankar, Integral solutions of ternary quintic Diophantine
quation
522
)12( zykx 
,International Journal of Mathematical Sciences 19(1-2), (jan-june 2010),165-169.
[6]. M.A.Gopalan,G.Sumathi & S.Vidhyalakshmi, Integral solutions of non-homogeneous quintic equation with three unknowns
5222
)3(1 zkyxxyyx
n

, International Journal of Innovativeresearch in Science, Engineering and
technology, Vol 2, Issue 4, April 2013, Pp 920-925.
[7]. S.Vidhyalakshmi ,K.Lakshmi and M.A.Gopalan , Integral solutions of non-homogeneous ternary quintic equation
0,,)(
522
 bazbabyax
,Archimedes J.math., 3(2),2013,Pp 197-204.
[8]. S.Vidhyalakshmi, K.Lakshmi and M.A.Gopalan , Integral solutions of non-homogeneous ternary quintic equation
0,,)(
522
 bazbabyax
,International journal of computational Engineering research, Vol 3, Issue 4, April
2013,Pp 45-50.
[9]. M.A.Gopalan,G.Sumathi & S.Vidhyalakshmi, Integral solutions of non-homogeneous ternary quintic equation in terms of pells
sequence
533
2)( zyxxyyx 
,JAMS, Vol.6, no.1,April 2013, page no.59-62