OBSERVATIONS ON THE HOMOGENEOUS QUINTIC EQUATION WITH FOUR UNKNOWNS 5 5 5 2 2 2 x y 2z 5(x y)(x y )w

IJESM Volume 2, Issue 2 ISSN: 2320-0294
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International Journal of Engineering, Science and Mathematics
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June
2013
OBSERVATIONS ON THE HOMOGENEOUS QUINTIC
EQUATION WITH FOUR UNKNOWNS
5 5 5 2 2 2
2 5( )( )x y z x y x y w
S.Vidhyalakshmi
K.Lakshmi*
M.A.Gopalan*
ABSTRACT
We obtain infinitely many non-zero integer quadruples ( , , , )x y z w satisfying the quintic
equation with four unknowns.
5 5 5 2 2 2
2 5( )( )x y z x y x y w .Various interesting properties
among the values of x, y, z and w are presented.
KEYWORDS: Quintic equation with four unknowns, integral solutions.
MSC 2000 Mathematics subject classification: 11D41.
NOTATIONS:
. ,
( 1)( 2)
1
2
m n
n m
T n -Polygonal number of rank n with size m
1
( 1)(( 2) (5 )
6
m
nP n n m n n - Pyramidal number of rank n with size m
( 1)pPR n n -Pronic number of rank n
6 ( 1) 1nS n n -Star number of rank n
1
2 ( 1)
3
n n
nJ -Jacobsthal number of rank n
2 ( 1)n n
nj - Jacobsthal-Lucas number of rank n
2
(2 1) 2n
nKY -keynea number.
Department of Mathematics, Shrimati Indira Gandhi College, Trichy

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1. 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 antiquity[1-3].For illustration, one may refer [4-5] for quintic
equation with three unknowns and [6-7] for quintic equation with five unknowns. This paper
concerns with the problem of determining non-trivial integral solutions of the homogeneous
quintic equation with four unknowns given by 5 5 5 2 2 2
2 5( )( )x y z x y x y w .A few
relations among the solutions are presented.
2. Method of Analysis:
The diophantine equation representing the quintic equation with four unknowns under
consideration is
5 5 5 2 2 2
2 5( )( )x y z x y x y w (1)
It is observed that (1) is satisfied by the following non-zero distinct integer quadruples:
2 2 2
( , , , ) :(6 , 2 ,4 ,3 ),(2(2 2 1),2(2 2 1),4 ,2 1)x y z w k k k k k k k k k k
However, we have other patterns of solutions which are illustrated below:
2.1: Pattern I:
Introduction of the transformations
, ,x u v y u v z v (2)
in (1) leads to
2 2 2
2 4.u v w (3)
which is of the form 2 2 2
z Dx y .
Using the most sited solution of the above equation, the corresponding non-zero
distinct integral solutions of (1) are given by
2 2
2 2
2 2
2 4 4
2 4 4
4
2
x p q pq
y p q pq
z pq
w p q
(4)
Following are some interesting relations between the solutions of (1):

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2013
1. 5, 4,( , ) ( , ) ( , ) ( , ) 2 p p px p p y p p z p p w p p T PR T
2. Each of the following expression is a nasty number:
(a)6[4 ( , ) ( , ) ( , )]w p q x p q y p q .
(b).6[ ( ,1) ( ,1) ( ,1) ( ,1)] 12x p y p z p w p
3. 4, 10, 4,( , ) ( , ) ( , ) 2 8 1p p q qx p q z p q w p q T S T t
4. 3, 6, 4,( , 1) ( , 1) ( , 1) 2 ( , 1) 6 2 2p P Px p p y p p w p p z p p T T T
5. 2 2 4
3, 4, 5, 4,( ( 1), ) ( ( 1), ) 8 6 2 3 0p p p p pZ p p p W p p p T T P T T
6. 2
2 2 2(2 ,1) (2 ,1)n n
n nz w j j
7. 2(2 ,1) (2 ,1) 4 4n n
n nx y KY j
8. The triple ( ( , ), ( , ), ( , ))x p p y p p z p p satisfies the homogeneous cone 2 2 2
2Y X Z
2.2: Pattern II:
In (2), the choice 2 , 2v V u U (5)
gives
2 2 2
2U V w (6)
After performing a few calculations, the integral solutions of (1) are obtained as
2 2
2 2
2 2
4 2 4
4 2 4
4
2
x p q pq
y p q pq
z pq
w p q
(7)
Note: Replacing q by p and p by q , x by x and y by y , in (7), we obtain the solutions of
pattern (1).
The above solution set (7) satisfies the following properties:
1. 3, 6, 4,( , ) ( , ) ( , ) ( , ) 22 11 22p p px p p y p p z p p w p p T T T
2. Each of the following expression is a nasty number:
(a).3[4 ( , ) ( , ) ( , )]w p q x p q y p q .
(b). 2 1(2 ,1) (2 ,1) (2 ,1) 6n n n
nx y z j
3. 3, 5, 4,( 2, 1) ( 2, 1) 16 32 48p P Px p p y p p T T t is a biquadratic integer.

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4. 3,20 ( ,1) ( ,1) ( ,1) ( ,1) 0(mod3)pT x p y p z p w p
5. 2(2 ,1) (2 ,1) (2 ,1) (2 ,1) 42 0(mod5)n n n n
nx y z w J
6. The triple ( ( , ), ( , ), ( , ))x p p y p p z p p satisfies the homogeneous cone 2 2 2
2X Z Y
2
5 4
3, 4,4,
7. ( ( 1), ) ( ( 1), ) ( ( 1), ) ( ( 1), ) 8 2 12 6p p p p pp
x p p p y p p p z p p p w p p p P T P T PR T
In addition to the above two patterns, there are two more patterns of solutions (1) which we
present below.
2.3: Pattern III:
Assume 2 2
2 , , 0w p q p q (8)
Write 4 as
2
(2 4 2)(2 4 2)
4
3
i i
(9)
Using (8) & (9) in (3) and applying the method of factorization define:
2
(2 4 2)( 2 )
( 2 )
3
i p i q
u i v
Equating real and imaginary parts, we get
2 2
2 2
2
( 2 8 )
3
4
( 2 )
3
u p q pq
v p q pq
(10)
In view of (2) and (10) the solutions of (1) are obtained as
2 2
2 2
2 2
2 2
18( 2 2 )
6( 2 10 )
12( 2 )
9( 2 )
x p q pq
y p q pq
z p q pq
w p q
(11)
2.4: Pattern IV:
Consider (6) as
2 2 2
2 1U V w (12)
Take 1 as

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2013
2
(7 4 2)(1 4 2)
1
9
i i
(13)
Using (8) & (13) in (12) and applying the method of factorization define:
2
(7 4 2)( 2 )
( 2 )
9
i p i q
U i V (14)
Equating real and imaginary parts, we get
2 2
2 2
1
(7( 2 ) 16 )
9
1
(4( 2 ) 14 )
9
U p q pq
V p q pq
(15)
In view of (2), (5) and (15) the integral solutions of (1) are found to be
2 2
2 2
2 2
2 2
2(11 22 2 )
2(3 6 30 )
2(4 8 14 )
9( 2 )
x p q pq
y p q pq
z p q pq
w p q
(16)
3. Conclusion:
It is to be noted that, instead of (13), one may write 1 as
2
(1 2 2)(1 2 2)
1
3
i i
Following the procedure presented above, the corresponding integral solutions of (1) are
obtained.

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2013
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 Dio.problem 3 3 5
2x y z ,
Advances in Mathematics, Scientific Developments and Engineering Application,
Narosa Publishing House, Pp 1-6, 2010.
5 .M.A.Gopalan & A.Vijayashankar, Integrated solutions of ternary quintic
Diophantine equation 2 2 5
(2 1)x k y z ,International Journal of Mathematical
Sciences 19(1-2), 165-169,(jan-june 2010)
6. M.A.Gopalan & A.Vijayashankar, Integrated solutions of non-homogeneous quintic
equation with five unknowns 5
xy zw R , Bessel J.Math.,1(1),23-30,2011.
7. M.A.Gopalan & A.Vijayashankar, solutions of quintic equation with five
unknowns 4 4 2 2 3
2( )x y z w P ,Accepted for Publication in International Review
of Pure and Applied Mathematics.

OBSERVATIONS ON THE HOMOGENEOUS QUINTIC EQUATION WITH FOUR UNKNOWNS 5 5 5 2 2 2 x y 2z 5(x y)(x y )w

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