# Is anything known about which numbers appear in the continued fraction expansion of $\pi$?

This question is mostly idle curiosity, and certainly is not related to any research activities of my own. The motivation and background are as follows. I am currently teaching a Freshman Seminar in which one of the main texts is Hofstadter's book Gödel, Escher, Bach. In one of its dialogs, the Tortoise explains that he has found a counterexample to Fermat's Last Theorem, and the exponent $n$ in question is the unique positive integer not appearing in the continued fraction expansion of $\pi$ — and this is enough information for you also to find it.

This is funniest if the following were true at the time of GEB's writing (1978):

(a) it was widely believed, but not known, that every positive integer appears somewhere in the continued fraction expansion of $\pi$;

(b) proving this was far beyond the technology of the time, and even saying anything useful about which numbers do and don't appear (beyond calculating the first few numbers that do appear) looks interminable.

Based on what little I know about similar problems, my guess is that (a) and (b) are both true, and remain true today. Evidence for this impression is OEIS sequence A225802. But this is far from my area of expertise. Thus, my questions are:

1. Were (a) and (b) true in 1978?
2. Do they remain true today?
• Hopefully someone can expand on this (to make it more suitable as a formal answer), but the quick version is yes: (a) and (b) were true in 1978 and remain true today. – Henry Cohn Apr 23 '14 at 3:40
• It is conjectured that the geometric mean of these integer-valued coefficients is Kinchin's Constant, ~ 2.68545 ... the conjecture is well-grounded in ergodic theory – John Sidles Apr 23 '14 at 4:22
• That's an intense freshman seminar! – Nate Eldredge Apr 23 '14 at 5:59
• @NateEldredge Well, we're not actually going into much of the mathematics, but I want to be able to give a correct answer if asked. – Theo Johnson-Freyd Apr 23 '14 at 14:48
• Just having GEB on the reading list puts it in the "intense" category as far as I'm concerned :) – Nate Eldredge Apr 23 '14 at 16:02

The Gauss-Kuzmin Theorem says that if $x$ is chosen uniformly at random from (say) $[0,1)$ (thanks, John Bentin) then as $n\to\infty$ the probability that the $n$th partial quotient of $x$ is $k$ tends to $$\log_2{(k+1)^2\over k(k+2)}$$ It was widely believed in 1978, and is still widely believed today, that $\pi$ acts, in this regard, like a random number. We are 36 years closer to proving this today than we were in 1978.
• Statistics are the same for $[0,1)$ and $[n,n+1)$. – Alexey Ustinov Apr 24 '14 at 0:10