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 notations 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 >=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 Chebychev 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 split it into two subcases; 239^2+1=2*13^4 killed one of them.)
[Obvious note: I would give the exact MAA reference...if I could.]