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


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?

  • 2
    $\begingroup$ 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 '19 at 3:41

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