Find all pear (x,y) where x,y integer number and x not equal y, such that :
(x^2+2y^2)/(2x^2+y^2) is a perfect square number ?

Question

Find all pear (x,y) where x,y integer number and x not equal y, such that : 
(x^2+2y^2)/(2x^2+y^2) is a perfect square number ?

anonymous · Answer

So my brain has stopped working or this is trivial !!!
$$\frac{x^2+2y^2}{2x^2+y^2}=n^2$$$$x^2+2y^2=(2x^2+y^2)n^2=2x^2n^2+y^2n^2$$$$x^2-2x^2n^2=y^2n^2-2y^2$$$$x^2(1-2n^2)=y^2(n^2-2)$$for  $n>1$  LHS is negative and RHS is positive...so  $n=1$ and  $y=\pm x$

anonymous · Answer

@mukushla from this how can we find all possible pairs??

anonymous · Answer

well i proved that $x=\pm y$  so all pairs are in the form $(x,y)=(\pm a,\pm a)$

experimentx · Answer

x not equal y ,,, so (+a, -a) ..lol

anonymous · Answer

lol

experimentx · Answer

when i comes to Diophantine eqn ... i haven't found anyone better than you man!!

anonymous · Answer

@RadEn 

what do u think?

raden · Answer

so, there are infinity pairs, isn't mukushlah?

anonymous · Answer

yeah solutions are infinite

raden · Answer

Ok, thank u...
sorry, my conection very low...
i come late see it's answer :(

anonymous · Answer

:)

raden · Answer

wait, dont run every where.... 
i have one more, iwll repost soon ... :D

anonymous · Answer

ok
@experimentX
Oh santosh i see ur last reply now...that means a lot from u man...thank u

experimentx · Answer

no probs man!!