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Champernowne's constant is an artificially-constructed mathematical constant defined by concatenating all the positive integers' representations:

$C_{10}=0.1234567891011121314151617181920212223242526272829\ldots$

Its continued fraction displays extremely large numbers at locations:

$M=\left\{5, 19, 41, 102, 163, 247, 358, 460, \ldots\right\}$

And the next incrementally largest term greater than or equal to $10^5$ occurs at positions:

$P=\left\{4, 18, 40, 162, 526, 1708, 4838, 13522, 34062,\ldots\right\}$

My guess is that the set of all large terms could be written as $\bigcup^\infty_{n=0}\left(\text{something}\right)$. My question is this: Is there a way (formula) to determine where these large numbers occur and their specific values? What about determining the locations of only the high-water marks?

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    $\begingroup$ You've probably already seen this paper and this paper. There are some conjectures but it seems that almost nothing is actually known. $\endgroup$
    – Mr. G
    Jun 9, 2019 at 3:41
  • $\begingroup$ I don't have an answer, unfortunately, but, as pointed out in the Wikipedia article, the convergents obtained by truncating just before the large numbers will be especially good rational approximations of $C_{10}$. This suggests a different question: what is the nature of the set of good rational approximations to $C_{10}$ that gives rise to these large numbers? The Wikipedia article describes the set of rational approximations that appears to coincide with the high-water marks. I've recently edited the article to make this more explicit. $\endgroup$
    – Will Orrick
    Sep 4, 2022 at 13:17
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    $\begingroup$ As for the location and value of the large numbers, that seems likely to me to be a complicated question, as it will depend on the number-theoretic properties of the rational approximations. The paper On the length of continued fractions representing a rational number with given denominator by Peter Szüsz may provide some insight into the general issues. $\endgroup$
    – Will Orrick
    Sep 4, 2022 at 13:21

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