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Integral solution of the biquadratic Equation
II. METHOD OF ANALYSIS
The non-homogeneous biquadratic diophantine equation with four unknowns is
4 4
2
2 2
x y ( k 1)( z w )
(1)
It is worth to note that (1) is satisfied by the following non-zero distinct integer quadruples
2 2
2 2
2 2 2
2 2 2
(( a b )( 1 k ), ( a b )( 1 k ), ( a b ) ( 2 k 1), ( a b ) ( 2 k 1)) and
2
2
4
4
(10 b ( k 1), 10 b ( k 1), 100 b ( 2 k 1), 100 b ( 2 k 1)) .
However we have some more patterns of solutions to (1) which are illustrated below
To start with,
the substitution of the transformations
2
2
x u v , y u v , z 2 uv , w 2 uv
(2)
2 2
2
2
in (1), leads to u v ( k 1)
(3)
2.1 Pattern 1
2 2
Let a b
(4)
Substituting (4) in (3) and using the method of factorization, define
2
iv ( k i )( a ib )
(5)
u
Equating real and imaginary parts in (5), we have
2
2
u k ( a b ) 2 ab
(6)
2 2
v ( a b ) 2 abk
Substituting (4) , (6) in (2) and simplifying, the corresponding values of x,y,z and w are represented by
x ( k , a , b )
2 2
2 2
k (a b 2 ab ) (a b 2 ab )
y ( k , a , b )
2
k (a
2
b
2
2 ab ) (a
2
b
2 ab )
z ( k , a , b )
2 2
4 k ab (a
2
4
b ) 2 k (a
4
b
2 2
2
6 a b ) 4 ab (a
2 2
b ) (a
2 2
b )
Properties
w ( k , a , b )
2 2 2
4 4 2 2
2 2 2 2 2
4 k ab (a b ) 2 k (a b 6 a b ) 4 ab (a b ) (a b )
4
1. 4 z ( k , a 1, a ) w ( k , a 1, a ) 192 PT 96 T 24 PR
is a cubical integer.
2.Each of the following is a nasty number:
(i) 3(z(k,a,b)-w(k,a,b))
(ii) x(k,ka,a)
(iii) -6(x(k,a,a)+y(k,a,a)).
a
2
3
5
3. z k , a , b ) w ( k , a , b ) 48 ( k 1) P k (32 P 48 F 8t
4) 0(mod 28 )
( a 1
a
4 , a , 4 9 , a
n
n
4. x k ,2 ,1) y ( k ,2 ,1) 12 kJ KY j o (mod )
( k
n
2 n 2 n
3
2
5. x k ,1, b ) y ( k ,1, b ) 48 kP ( k 1)( 24 PT 9OH
t 18 t 36 t 1)
(
b 1 b
b 19 , b 6 , b
4 , b
2.2 Pattern2:
(3) is written as
Write „1‟ as
a
2 2
2
2
u v ( k 1) * 1
(7)
( 3 4 i )( 3 4 i )
1=
25
Using (4) and (8) in (7) and employing the method of factorization, define
(3 4 i )
u
5
Equating real and imaginary parts of (9) we get
2
iv ( k i ) ( a ib )
(9)
a
(8)
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