Download - ijcer

Integral Solutions of the Homogeneous Cubic Equation
The corresponding integral values of
x , y,
z,
w and X satisfying (1) are obtained as,
x(
u,
a.
b)
u a
y(
u,
a,
b)
u a
2
2
b
2
b
2
2 2 n
n
z(
u,
a,
b)
u ( a b ) cos 2absin
2 2
2 2 n
n
w(
u,
a,
b)
u ( a b ) cos 2absin
2 2
2 2 n
n
X ( a,
b)
( a b ) sin 2abcos
2
2
Properties:
1.Each of the following expression is a nasty number
2 n
2
3 x(
u,
a,
b)
y(
u,
a,
b)
( 1)
12 x(
u,
a,
b)
3( z(
u,
a,
b)
w(
u,
a,
b))
2
i.
2 2
6 x(
u,
a,
b)
y(
u,
a,
b)
( a b
ii.
2. ( 2x(
u,
a,
b),
z(
u,
a,
b)
w(
u,
a,
b),
x(
u,
a,
b)
y(
u,
a,
b))
forms a Pythagorean triple.
3.If
)
2
a, b are taken as the generators of the Pythagorean triangle ( , ,
) whose sides are
2 2
2 2
a b ; 2ab;
a b then the product
n
z(
u,
a,
b)
w(
u,
a,
b)
n
n
z(
u,
a,
b)
w(
u,
a,
b)
X
( a,
b)sin
cos
X
( a,
b) cos
sin
2 2 2
2 2
represents two times its area.
4. x ( u,
a,
b)
y(
u,
a,
b)
z(
u,
a,
b)
w(
u,
a,
b)
0(mod2)
5. x( u,
a,
b)
y(
u,
a,
b)
0(mod2)
2.3Pattern 3:
In (6) 1 can be written as
Proceeding as in Pattern II
p
2 2
2 2
( p q i2
pq)(
p q i2
pq)
1 ; p q 0
2 2 2
( p q )
( p
q
i2
pq)(
p q
( p q )
i2
pq)
2 2
2 2
2
iX ( a ib)
2 2 2
(10)
Equating real and imaginary parts,
X
p
( p
)
( p
q ) pq
4ab
q ) ( p q
2 2
( 2 2
a b
2 2
2 2
(11)
2pq
)
2
( p q
( p
2ab
) ( p
q
2 2
2 2
( a b
2
2 2
q
www.ijceronline.com ||May ||2013|| Page 25
)
)
(12)
)
n
2