Prove that there does not exist any pair (x, y) of positive integers such that: 4xy - x – y is a perfect square.

the given expression=4xy-x-y

=(x+y)^2-(x-y)^2-(x+y)

=(x+y)(x+y-1)-(x-y)^2

now,the first part cant be a perfect square.so,

to make expression a perfect square,(x+y)(x+y-1)=0

so,(x+y)=0

or,(x+y)=1.

so,it is clear,that,both x & y cannot be positive integers

Posted by subhobrata on 2009-11-02 02:08:26