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?



Infinitely many ones in continued fraction of pi?
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.


