I never realized that integer comparison is so much cheaper than rationals comparison. Using these conditions one can speed up the procedure. Since both expressions are even, we know they at least share 2 as a common factor. We can always get lucky and obtain even more common factors and make a leap, but halving each time we double n is good enough. I don't know if these comments apply.
Factoring is very time consuming.
This is a times savings for moderate 'n'. Instead of exhaustive search, I decided to quickly check special cases. Here is some simple Maple code to play with.