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Integral Solutions of the Homogeneous Cubic Equation
which is satisfied by
v m
v m
2
2
2
2 2
n , p 2mn,
X m n
m
n o
2
2 2
n , p m n , X 2mn
(5)
Substituting (5) in (2) , the two sets of solutions satisfying (1) are obtained as follows:
SET 1:
x(
u,
m,
n)
u m
y(
u,
m,
n)
u m
z(
u,
m,
n)
u 2mn
w(
u,
m,
n)
u 2mn
X ( m,
n)
m
2
n
2
2
2
n
n
2
2
SET 2:
x(
u,
m,
n)
u m
y(
u,
m,
n)
u m
z(
u,
m,
n)
u m
w(
u,
m,
n)
u m
2
2
2
2
n
n
n
2
2
2
n
2
X ( m,
n)
2mn
2.2Pattern 2:
Equation (3) can be written as
2 2 2
p X v 1
(6)
Assume
v
2 2
a b
(7)
Write 1 as
2n
2n
(1 i)
(1 i)
1 (8)
2n
2
Substituting (7) and (8) in (6) and using the method of factorization, define
2n
2 (1 i)
p iX ( a ib)
(9)
n
2
Equating real and imaginary parts of (9),we have
p ( a
X ( a
n
n
b )cos 2absin
2 2
2 2
n
n
b )sin 2abcos
2 2
2 2
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