I noticed in the Continued Fraction expansion of the Euler-Mascheroni Constant that some numbers recur a lot, like the number 1 or the number 10. Is it known if there are infinitely many of the same number in the Continued Fraction expansion of the Euler-Mascheroni Constant? In particular, does 1 appear infinitely many times?
$\begingroup$
$\endgroup$
5
asked Mar 4, 2023 at 22:20
-
5$\begingroup$ Since it is not known whether the E-M constant is rational or not, of course your answer is unknown. $\endgroup$– Gerald EdgarCommented Mar 4, 2023 at 22:29
-
$\begingroup$ I believe that for almost all reals, every positive integer appears infinitely many times in the continued fraction expansion. Assuming a vague conjecture that Euler-Mascheroni constant is "generic" when it comes to the continued fraction expansion, it should have that property too. $\endgroup$– WojowuCommented Mar 4, 2023 at 23:06
-
1$\begingroup$ Moreover, for almost all reals, $1$ appears with density about $41.5\%$ among the partial quotients in the continued fraction expansion, so it's not surprising there are a lot of ones in the expansion of $\gamma$. See en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribution $\endgroup$– Gerry MyersonCommented Mar 4, 2023 at 23:43
-
$\begingroup$ @GeraldEdgar That logic doesn't make sense. It could be known that the E-M constant does not have $1$ infinitely often in its continued fraction expansion (though I of course doubt it). $\endgroup$– mathworker21Commented Mar 4, 2023 at 23:44
-
1$\begingroup$ @mathworker21 The logic is that if someone had proved such an astounding result, it would be headline news, and we would all have heard about it. $\endgroup$– Timothy ChowCommented Mar 5, 2023 at 4:23
Add a comment
|