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When we work with rational numbers, our continued fraction will have a finite number of elements.

Are there ways to estimate the number of elements of a continued fraction when expanding a rational number? For example, in the following form: "For the chosen rational number N, continued fraction will contain at least/at most/strictly M elements"

The obvious way is to implement the continued fraction decomposition algorithm and simply count the number of elements each time the loop passes.

But this method can be quite time-consuming if there are a large number of elements of the continued fraction. I tried to find the answer and the only thing I found was this thread: Nth number of continued fraction

But it ultimately came down to what I said above.

dtn
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    The upper bound will be a constant times $\log d$ where $d$ is the denominator of the rational number; it will be asymptotically attained when all the convergents equal $1$, which happens when the numerator and denominator are consecutive Fibonacci numbers. Statistical results on the number of convergents are also available (somewhere). This turns out to be the same question as the number of steps needed in the Euclidean algorithm to calculate the greatest common divisor of the numerator and denominator. – Greg Martin Oct 16 '23 at 07:05
  • @GregMartin https://math.stackexchange.com/questions/2320314/continued-fraction-for-sqrtd-upper-bound-of-cycle-length https://mathoverflow.net/questions/264315/cycle-length-and-total-count-of-gauss-reduced-indefinite-binary-quadratic-forms Isn't this from here by any chance? – dtn Oct 16 '23 at 15:16
  • I think that's a different topic: that post talks about the period lengths of infinite continued fractions, while this post asks about the lengths of terminating continued fractions. – Greg Martin Oct 16 '23 at 16:30
  • @GregMartin https://www.mathnet.ru/links/ae17de6c2589d21ac181d3f35450b77a/sm8718_eng.pdf http://matwbn.icm.edu.pl/ksiazki/aa/aa37/aa3717.pdf In these works, it’s the same thing you talked about - there is a certain constant in the inequalities. And if we can estimate the numerator/denominator, then it is difficult to estimate this constant. – dtn Oct 16 '23 at 16:33

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