Download - ijcer

Integral Solutions of the Homogeneous Cubic Equation
Since our aim is to find the integral solutions, substituting
and (12)
2 2
2 2
a ( p q ) A,
b ( p q ) B
in (7),(11)
2 2 2 2 2
v ( p q ) ( A B )
(13)
2 2 2 2 2 2
p ( p q ) ( A B )( p q ) 4ABpq
(14)
X
2 2 2 2
2 2 2 2
2 pq(
p q )( A B ) 2( p q ) AB(
p q ) (15)
Substituting (13), and (14) in (2) and using (15) we have the integral solutions of (1) as,
Properties:
x(
u,
A,
B)
u ( p
y(
u,
A,
B)
u ( p
z(
u,
A,
B)
u ( p
w(
u,
A,
B)
u ( p
X ( A,
B)
2 pq(
p
2
2
2
2
2
q
q
q
q
2
2
2
2
q
)
)
)( A
2
2
2
)( A
)( A
( A
( A
2
2
2
2
2
B
B
B
B
2
2
B
)
)
)( p
2
2
2
2
)( p
2
) 2( p
q
2
2
q
2 2
1. 4 pq z(
u,
p,
q)
w(
u,
p,
q)
2( p q ) x(
u,
p,
q)
0(mod 8)
4 4
2. 2 ( p q ) w(
u,
p,
p 1)
z(
u,
p,
p 1)
0(mod8)
G p
) ( p
2
q
) ( p
2
2
2
q
) AB(
p
2
2
q
)4ABpq
2
)4ABpq
2 2
2 2 3
3. pq z
u,
p,
p 1)
w(
u,
p,
p 1)
( p q ) x(
u,
p,
p 1)
2( p q ) 0
q
( P p 1
2 2 2
4. 2( q ) S 3y(
u,
p,
p 1)
x(
u,
p,
p 1)
0(mod4)
p
p
5. 3 z(
u,
p,
p)
w(
u,
p,
p)
pq 0(mod 2)
6.
OH pq
w( u,
p,
q)
q
2 2 2
z(
u,
p,
q)
x(
u,
p,
q)
10( p q ) is a perfect square.
7. Each of the following expression is a nasty number
i. 6ux(
u,
A,
B)
y(
u,
A,
B)
z(
u,
A,
B)
w(
u,
A,
B)
ii. 6x(
u,
A,
A)
y(
u,
A,
A)
III. 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 2,Chelsea Publishing Company, New York, 1952.
[2]. L.J .Mordell, Diophantine Equations, Academic Press,London,1969.
[3]. M.A. Gopalan, and S. Premalatha, Integral solutions of ( x
2
2
y)(
xy w ) 2( k
3
1) z , Bulletin of Pure
and Applied Sciences, Vol.29E,No.2,197-202, 2009.
[4]. M.A. Gopalan, and V. Pandichelvi, Remarkable Solutions on the cubic equation with four
3 3 3
2
unknowns x y z 28(
x y z)
w , Antarctica J Math, ,Vol.7, No.4,393-401, 2010.
[5]. M.A. Gopalan, and B.Sivakami, Integral solutions of homogeneous cubic equation with four unknowns
3 3 3
3
, Impact. J.Sci.Tech.Vol.4,No.3,53-60,2010.
x
y
z
3xyz
2( x y)
w
2
)
www.ijceronline.com ||May ||2013|| Page 26