-------- Original Message -------- Subject: Re: [math-fun] [EXTERNAL] Re: Nines in Pi Date: 2014-04-11 16:36 From: "Schroeppel, Richard" <rschroe@sandia.gov> To: math-fun <math-fun@mailman.xmission.com> Cc: "rcs@xmission.com" <rcs@xmission.com>, "Schroeppel, Richard" < rschroe@sandia.gov> Reply-To: math-fun <math-fun@mailman.xmission.com> IIRC, the definition of "irrationality measure" M for X uses the expression | X - n/d | > 1/d^M except for a finite number of ds. So the digits n through 7.61n (of pi) can't be all 9s (or all 0s), with only a finite number of exceptional n. <RCS So there are still probably millions of cases of a million 9s in a row. Man, we don't know anything. --rwg RCS>At a symbolic math conference in NYC ~1985, the Chudnovsky's offered a good result for pi^2, perhaps 6 or 7, and remarked that that implied a less good result for pi, perhaps 12-14. Note that the ordinary continued fraction algorithm gives an infinite stream of d when M=2, for any irrational X. So we'd need something a little stronger to address Jim's original question. And (Thue-Seigel-)Roth says that M<2+epsilon for (irrational, real) algebraic X. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [ math-fun-bounces@mailman.xmission.com] on behalf of Michael Kleber [ michael.kleber@gmail.com] Sent: Thursday, March 20, 2014 8:20 AM To: math-fun Subject: [EXTERNAL] Re: [math-fun] Nines in Pi The irrationality measure of pi is at most 7.6063 [Salikhov, "On the Irrationality Measure of $\pi$", 2008]. I think that translates into "There are at most finitely many n for which digits n through k*n are all 9's" for some k, but I don't know what k is. But none of this answers Jim's / Dick's question (whose answer is surely "it never happens but we don't have a proof"). --Michael On Thu, Mar 20, 2014 at 9:34 AM, James Propp <jamespropp@gmail.com> wrote:
Is it possible that for some n, the n+1st through 2nth digits of pi are all 9's?
Dick Hess just gave a wonderful G4G talk which raised this issue.
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.