[math-fun] Nines in Pi
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
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.
Just curious: Can one calculate the exact probability of this (assuming for simplicity that the digits of pi are independent and random)? Or better: Just take flips of a fair coin (H or T*) indexed by Z+. What is the probability there exists an n such that flips n+1 through 2n are all H ??? Dealing with the overlaps looks complicated. —Dan _____________________________________________________________________________ * If the coin were an old French franc, would a flip be either a T or a Q ??? On Mar 20, 2014, at 7:20 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
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. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On the contrary, it always happens for any number and base, at n = 0 ! If on the other hand there is prior knowledge constraining n from below --- as in the case of pi, for which the answer is presumably known when n < 10^12 or whatever --- then the probability for a normal number becomes correspondingly exponentially smaller. WFL On 3/20/14, Dan Asimov <dasimov@earthlink.net> wrote:
Just curious: Can one calculate the exact probability of this (assuming for simplicity that the digits of pi are independent and random)?
Or better: Just take flips of a fair coin (H or T*) indexed by Z+. What is the probability there exists an n such that flips n+1 through 2n are all H ??? Dealing with the overlaps looks complicated.
--Dan _____________________________________________________________________________ * If the coin were an old French franc, would a flip be either a T or a Q ???
On Mar 20, 2014, at 7:20 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
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. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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. 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. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (5)
-
Dan Asimov -
Fred Lunnon -
James Propp -
Michael Kleber -
Schroeppel, Richard