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

(In reply to No Subject by subhobrata)

I noticed a slight problem with your proof.
you state that for (x+y)(x+y-1)-(x-y)^2 to be a perfect
sqaure then (x+y)(x+y-1)=0 but a quick counter proof to this
is if (x+y)(x+y-1)=20 and x-y=4 then you would have
20-16=4=2^2
now of course this example does not lead to positive integer x,y solutions, but it does show that your conclusion is not always true.  Although I do agree that (x+y)(x+y-1) can never be a perfect square

Posted by Daniel on 2009-11-02 12:33:50