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(2) n n
3n
n
x (2 ,1) y(2
,1) 3[ j 27J
j ( 1)
9( 1)
2]
3n
n 2n
(3) Each of the following represents a nasty number:
(a) 2 2
5
y (3a
,1) x(3a
,1) t 2 6 2 3
14,3a P
3 a
7{42
2x(
a,1)
y(
a,1
(b) )}
(c) 5
6{ x(
a,
a)
z(
a,
a)
16P a }
(d) 5
2{32 y(
a,
a)
z(
a,
a)}
P a
2.5Pattern:5
Instead of (15) we write 7 as
Observation on the Ternary Cubic Equation
(8 i2
12 )(8 i2
12 )
7
16
Following the procedure as presented in pattern4, the corresponding non-zero distinct integral solutions to (1)
are
3 2 2 3
x(
a,
b)
( a 36ab
) 10(3a
b 12b
)
3 2 2 3
y(
a,
b)
2(
a 36ab
) 8(3a
b 12b
)
2 2
z( a,
b)
a 12b
Properties:
(1) 24
x ( 1, n)
y(1,
n)
6CPn
t24,
n 2t6,
n 93t
4,
n 3
(2) 3
x ( n,1)
2CPn
2CP29,
n 16t3,
n 7t4,
n 122
3
(3) y ( n,1)
2[2CPn
74t3,
n 37t4,
n ] 8[ t8,
n 4t3,
n 2t4,
n 12]
(4) Each of the following represents a nasty number:
(a) 5
7{110P n x(
n,
n)
z(
n,
n)}
(b) 5
6{ x(
n,
n)
y(
n,
n)
z(
n,
n)
174P n }
2.6Pattern:6
(1) can be written as
2 2 3
( 2x
y)
3y
28z
(16)
One may write 28 as
28 (5 i 3)(5 i 3)
(17)
Substituting (17) and (4) in (16), employing the method of factorization, we have
3
( 2x
y)
i 3y
(5 i 3)( a i 3b)
Equating real and imaginary parts, we have
3 2 2 3
x ( a,
b)
3( a 9ab
) 3( a b b )
(18)
3 2 2 3
y(
a,
b)
( a 9ab
) 15(
a b b )
Thus (14) and (18) represents the non-zero distinct integral solutions to (1)
Properties:
(1) 9
x n,1)
2CP
t 48t
24t
3
( n 8, n 3, n 4,
n
(2) 13 9 3
y 1, n)
6CP
2CP
2CP
t 6t
2t
1
( n n n 20, n 3, n 4,
n
(3) 7
y (1, n)
18Pn
1(mod9)
(4) 5
x ( n,1)
6Pn
3 0(mod27)
(5) 3
P y(
n,1)
3t
15(mod2)
6 n
10,
n
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