# Lengths of continued fractions for the numbers with fixed ratio

Let $s(x)$ is the length of continued fraction expansion of $x$, and let $l(x)$ be the sum of partial quotients. I can prove that for any rational $\alpha$ ratios $\frac{s(\alpha x)}{s(x)}$ and $\frac{l(\alpha x)}{l(x)}$ (for all rational $x$) are bounded with some constants depending on $\alpha$ only.

Is this result new?

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I would be really surprised if it wasn't. After all, for l(x) this is just the length of the Euclid algorithm for x=p/q, so if we, say, multiply it by 2/3, then it is just the Euclid algorithm for 2p/3q... must be well known. Perhaps, even Euclid himself knew it. ;) –  Nikita Sidorov Oct 9 '10 at 10:49
Yes, it is more or less clear. It is not surprising. It is not hard. But was it already proved or not? –  Alexey Ustinov Oct 9 '10 at 12:07
@Nikita It is not Euclid algorithm for 2p/3q at all :), and all the proofs I know are quite non-trivial, though some of them are respectively short. –  Fedor Petrov Oct 9 '10 at 13:10