<s>[INST] If x and y are different integers , and if $2005 +x =y^2 ; 2005+y =x^2 $ then find xy... Problem : 
If $2005 +x =y^2 ; 2005+y =x^2$ then find xy...
My approach : 
Let $2005 +x =y^2 .....(i) ; 2005+y =x^2 ......(ii) $
Now from (i) we get : 
$ y = \sqrt{x + 2005}$ 
Now putting this value of y in (ii) we get : 
$ \Rightarrow 2005 +\sqrt{x + 2005} =x^2$ 
$ \Rightarrow \sqrt{x + 2005} =x^2 -2005 $ 
Now squaring both sides we  get : 
$\Rightarrow (\sqrt{x + 2005})^2 =(x^2 - 2005)^2$ 
Is there any other way I can solve this problem please suggest... thanks.
 [/INST] $2005 +x =y^2 .....(i) ; 2005+y =x^2 ......(ii)$
$(i)-(ii)\Leftrightarrow (x-y).(x+y+1)=0$
$\Rightarrow x=y ; x+y=-1$
$+x=y$, now we solve $x,y$ then $xy=...$
$+x+y=-1 \Rightarrow xy=-2004$
</s>