Did you try Fermat's Infinite Descent ?
mmm , sorry after trying it , i don't think it's a good way to treat this exercice .
But we can also see that this equation is equivalent to this one ,
(because , as you said , x˛ is congruent to 0 (mod 2) so x is also congruent to 0 (mod 2) , then , x=2x' ...)
I also think that using inequalities is a good way ..... (i'll give it a try later )
P.S: Do we have to solve it in |N or Z ?