Here is a simply described but fiendishly diophanterrorizing problem I asked on AMM eons ago. Maybe you can shed some light upon it.
0.2 (base 4) = 0.2 (continued fraction)
0.24 (base 6) = 0.24 (continued fraction)
Find all examples of
0.$xyz$... (base B) = 0.$xyz$... (continued fraction).
First of all, both choices of notation define a rapidly closing interval nesting, and already on post-comma digit 2, you're down to one number by a simple > / < argument. But you may not use 0 for CF and $\ge B$ for base $B$, and thus almost any base $B$ will run into a dead end sooner or later. (It's fun to experiment with low $B$.)
Obvious Thing 1: 1-digit solution 0.$n$ for $B=n^2$.
Educated guess 2: There are only two solutions with two digits. (The second was listed in the MAA Answer Column; juggling with Chebyshev polynomials I had a sort of proof for that case, but it probably had more holes than a Menger sponge and so it wasn't printed there).
Wild guess 3: There is no solution with more than two digits, for the reasons above.
Can you at least prove case 2? (The MAA discussion splits it into two subcases; $239^2+1=2\times 13^4$ killed one of them.)
This is Monthly problem 10507. The problem appeared in February 1996 (volume 103, issue 2, page 173), and the discussion (and solution to other parts) in March 1998 (volume 105, issue 3, page 276). Thanks to Gerald Edgar for the pointer.