An interesting question from looking at the Stern-Brocot tree:
Two rational numbers, call them $a/b$ and $c/d$, are "adjacent" on the Stern-Brocot tree if and only if we have $ad-bc = \pm 1$, meaning that one will be parent to the other. (I think this also means that the continued fraction expansion of one is an initial segment of the other, except perhaps for the very last digit of that initial segment.)
Now suppose that we look at the set of all strictly positive rationals whose numerator and denominator are both $p$-smooth integers for some prime $p$. There will be infinitely many such rationals. Are there infinitely many "adjacent" $p$-smooth pairs if $p>2$? If not, how many are there and is there some way to compute them?
Better yet, suppose that we instead look at some other arbitrary finitely-generated subgroup of $\langle \Bbb Q^+, \cdot \rangle$, the multiplicative group of strictly positive rationals. This generalizes the situation with $p$-smooth numbers above. For which subgroups are there guaranteed to be infinitely many "adjacent" pairs?