Are there infinitely many ones in the simple continued fraction for pi? I know that there’s a probability distribution given through Gauss-Kuzmin, but is there a proof that there’s infinitely many ones? How about proofs that there are infinitely many of any positive integer in pi’s simple continued fraction?
$\begingroup$
$\endgroup$
1
-
3$\begingroup$ these are open problems, see math.stackexchange.com/q/2097946/87355 $\endgroup$– Carlo BeenakkerCommented Mar 12, 2023 at 15:21
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Nothing like that is known. There is likewise no proof that the decimal expansion of $\pi$ has infinitely many 1's (or any other specific digit).
The continued fraction expansions of algebraic numbers of degree greater than $2$ are equally opaque: none are known to have any specific positive integer appearing in them infinitely often. And the decimal expansion of no specific irrational algebraic number is proved to contain a specific digit infinitely often even though of course we expect each digit appears equally often.
These kinds of things are just hopeless to expect to be provable at present.
