I've just spent some time trying to find the proof, with little success, after noting

for some rational

, and then considering

as you said. I'm sure the proof is right under my nose, and I'm just not seeing it, do you have another hint? This is how far I got (granted, I could be "moving" in the wrong direction with these steps):

therefore

that leads to this equation:

this led me to:

But I feel like this is to complicated to be anywhere in the right direction, I did the isolation of the

and squared both sides in an attempt to get rid of irrationals so that I could use natural number factors to prove a common divisor between

and

.