On 9/12/13, Jeffrey Shallit <shallit@uwaterloo.ca> wrote:
This is related to the so-called "Zaremba conjecture"; you can find articles about it with that keyword.
It is known you can do it when B is a (sufficiently large) power of 2, 3, 5, and 6. The results are due to Niederreiter and others.
Jeffrey Shallit
--Oho, there is a largish literature on this. Shallit's tip led to http://arxiv.org/abs/1103.0422 http://arxiv.org/abs/1107.3776 where they claim that asymptotically 100% of integers B>0 have a matching A with 0<A<B, gcd(A,B)=1, such that the continued fraction of A/B has all partial quotients <=2189. It is suspected 2189 could be lowered to 5. Hensley conjectured if B is prime then it could be lowered to 2. Niederreiter 1986 showed for B any power of 2 or power of 3 there exists A such that continued fraction of A/B has all partial quotients <=3 or 4 (I do not have N's paper, this is from other people's conflicting descriptions of it). Harald Niederreiter: Dyadic fractions with small partial quotients, Monatsh. Math.,101,4(1986) 309-315. None of the papers I saw gave any substantial amount of computational evidence for anything. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)