JST Vol. 21 (1) Jan. 2013 - Pertanika Journal - Universiti Putra ...

Lemma 1.1
The equation
S. Ismail and K. A. Mohd Atan
4 4 3
x ± y = z has no solutions in the non-zero coprime integers, xyx. , ,
The following theorem gives the form of solutions to the equation
4 4 3
x + y = z in cases when gcd( abc , , ) > 1 and x¹ y in terms of n , where
4 4
æ x ö æ y ö
n = ç
÷ + ç ÷
ç
ç
çègcd( xyz , , )
÷ ø ç çègcd(
xyz , , )
÷ ø and
3k1 gcd( xyz , , ) n -
= , where k is a positive
integer with k ³ 1 .
Theorem 1.3:
Suppose the triplet ( abc , , ) with a¹ b and gcd = ( abc , , ) = d is a solution to the
equation,
4 4 3
x + y = z . Let
a b
u = , v = and
d d
3k1 integer. If d n -
3k1 = , then a un -
3k1 = , b vn -
4k1 = and c n -
= .
Proof:
Since a = du , b = dv and
From the following equation
d n -
3k-1 4 3k-1 4 3
( un ) + ( un ) = c or
Since,
4 4
n= u + v , we have
which implies that
4k1 c n -
=
3k1 = , we have clearly
4 4 3
a + b = c , it follows that:
( n ) c
4k- 1 3 3
=
4 4
n= u + v . Let k be a positive
a un -
( n )( u + v ) = c
3k-1 4 4 3
3k1 = and
3k1 Hence, a un -
3k1 = , b vn -
4k1 = and c n -
3k1 = if d n -
= , where
124 Pertanika J. Sci. & Technol. 21 (1): 283 - 298 (2013)
b vn -
3k1 = .
4 4
n= u + v
A corollary was obtained from the above theorem, as shown below. The forms of solution
for the triplet ( abc , , )
k
are obtainable when
c
w= = n .
d
Corollary 1.2
Suppose the triplet ( abc , , ) with a¹ b and gcd = ( abc , , ) = d is the solution to the
4 4 3 a b
equation x + y = z . Let u = , v = ,
d d
3k1 then a un -
3k1 = , b vn -
4k1 = and c n -
= .
Proof:
From the equation, we have
4 4 3
a + b = c .
4 4 4 3 3k
It follows that d( u + v) = dn .
c
w = and
d
4 4
n= u + v . If
k
w= n ,