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Groups > sci.physics.relativity > #674314
| From | Mild Shock <janburse@fastmail.fm> |
|---|---|
| Newsgroups | sci.physics.relativity, sci.physics |
| Subject | Superintelligenz ist auch F.U.D. (OAI:The-irrationality-exponent-of-pi-is-2-September-24-2026) |
| Date | 2026-10-10 00:35 +0200 |
| Message-ID | <11abq7c$17a99$3@solani.org> (permalink) |
| References | (12 earlier) <PTKdndjUXso-vLX3nZ2dnZfqnPudnZ2d@giganews.com> <-nGdnaezj6g3vrX3nZ2dnZfqnPudnZ2d@giganews.com> <bIycnWjd8J1CuLX3nZ2dnZfqnPadnZ2d@giganews.com> <11475ju$fsrh$1@solani.org> <119omsb$pkou$4@solani.org> |
Cross-posted to 2 groups.
Hi, An Elegant Continued Fraction for π L. J. Lange - Pages 456-458 | Volume 106, 1999 The American Mathematical Monthly https://www.tandfonline.com/doi/pdf/10.1080/00029890.1999.12005070 Meine Vermutung der jeweils zu einer bestimmten Tiefe berechnete Kettenbruch stimmt mit den partiellen Summen der Nilakantha Reihe überein. Aber ich weiss nicht ob man mit der Nilakantha Reihe so direkt rho=2 erhält, wie es den Anschein macht für den Kettenbruch. Bye See also this AI slop from a "frontier model": https://github.com/openai/math/tree/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026 https://github.com/openai/math/blob/main/reasoning_traces/irrationality-exponent-of-pi.pdf Mild Shock schrieb: > Hi, > > Now with integers, to really produce some results: > > % pi2(+Integer, +Rational, -Rational) > pi2(0, P/Q, R/Q) :- !, R is P-3*Q. > pi2(N, P/Q, S) :- > H is 6*P+(2*N-1)^2*Q, > M is N-1, > pi2(M, H/P, S). > > Now I see: > > ?- between(10,20,N), pi2(N,6/1,X), Y is X, write(N-X-Y), nl, fail; true. > 10-475113942765/151242416325-3.1414067184965018 > 11-11922290683065/3794809718700-3.1417360992606653 > 12-322869019821075/102776096548125-3.141479689004255 > 13-9388645795842075/2988412653476250-3.1416831892077552 > 14-291703390224616125/92854250304440625-3.1415189855952756 > 15-9646071455650881825/3070380543400170000-3.141653394197426 > 16-338203386739761387075/107655217802968460625-3.1415419859977827 > 17-12533792135642378629875/3989575718580595893750-3.1416353566793886 > 18-489501901570061970946125/155815096120119939628125-3.1415563302845726 > 19-20095772843114788169975625/6396619735457555416312500-3.1416238066678384 > 20-865107029346752986828909875/275374479611447760672253125-3.141565734658547 > true. > > And further: > > % roth2(+Integer, -Float) > roth2(N, X) :- > pi2(N, 6/1, A/_), > M is N+1, > pi2(M, 6/1, B/_), > X is 1+msb(B)/msb(A). > > Now I see, no more alternating: > > ?- between(10,20,N), roth2(N,X), write(N-X), nl, fail; true. > 10-2.1315789473684212 > 11-2.116279069767442 > 12-2.104166666666667 > 13-2.0943396226415096 > 14-2.086206896551724 > 15-2.079365079365079 > 16-2.0735294117647056 > 17-2.0684931506849313 > 18-2.0769230769230766 > 19-2.0595238095238093 > 20-2.056179775280899 > true. > > And for big values pretty much looks like 2+C/N: > > ?- roth2(1000,X). > X = 2.0011523151058035. > ?- roth2(10000,X). > X = 2.000116750597374. > ?- roth2(100000,X). > X = 2.0000111335844575. > > Bye > > BTW, because of the way I compute > the continued fraction, it takes some time: > > ?- time(roth2(100000,X)). > % Zeit 11258.941 ms, Benutzer 100 %, Lips 53 k > X = 2.0000111335844575. > > Mild Shock schrieb: >> Hi, >> >> Take this continued fraction: >> >> % pi(+Integer, +Float, -Float) >> pi(0, R, S) :- !, S is R-3. >> pi(N, R, S) :- >> H is 6+(2*N-1)^2/R, >> M is N-1, >> pi(M, H, S). >> >> Nice little corridor for pi: >> >> ?- between(10,20,N), pi(N,6,X), write(N-X), nl, fail; true. >> 10-3.1414067184965013 >> 11-3.1417360992606653 >> 12-3.141479689004255 >> 13-3.141683189207755 >> 14-3.141518985595275 >> 15-3.1416533941974265 >> 16-3.141541985997783 >> 17-3.141635356679389 >> 18-3.1415563302845726 >> 19-3.1416238066678384 >> 20-3.1415657346585473 >> true. >> >> We can als compute roth: >> >> % roth(+Integer, -Float) >> roth(N,X) :- >> pi(N, 6, A), >> M is N+1, >> pi(M, 6, B), >> X is 1+log(B)/log(A). >> >> Now we have a corridor for 2: >> >> ?- between(10,20,N), roth(N,X), write(N-X), nl, fail; true. >> 10-2.0000915947815723 >> 11-1.9999287043438398 >> 12-2.0000565883650765 >> 13-1.999954341912648 >> 14-2.000037375237465 >> 15-1.9999690217963195 >> 16-2.000025963563826 >> 17-1.9999780257693796 >> 18-2.000018763059092 >> 19-1.9999838523285578 >> 20-2.000013997167758 >> true. >> >> How proof limsup is 2? >> >> Proof Sketch: >> >> Lets check the formula 1+log(An+1)/log(An) . >> If An+1 < pi and An > pi. We know that >> 1+log(An+1)/log(An) < 1+log(pi)/log(An) >> and 1+log(An+1)/log(An) < 1+log(An+1)/log(pi). >> >> So we get limsup 1+log(An+1)/log(An) =< >> limsup 1+log(An+1)/log(pi) = 1+log(pi)/log(pi) = 2. >> But since pi is irrational the lowerbound >> >> is anyways 2, so that Liouvielle-Roth >> irrationality measure = 2 follows. >> >> Q.E.D. >> >> Bye
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Re: Hyperspectral Measurements for Inter-Calibration (Was: Nobody understands relativity) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-27 01:45 -0700
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Johnny Depp prevention [Windows 11 etc...] (Re: Homework: Game Engine in WebGPU) Mild Shock <janburse@fastmail.fm> - 2026-08-03 18:18 +0200
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Re: Hyperspectral Measurements for Inter-Calibration (Was: Nobody understands relativity) Mild Shock <janburse@fastmail.fm> - 2026-10-02 18:41 +0200
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Math is the premier AI Victim [Billions of $USD later] (Re: So it was a race against Buzzard. Well done! [Langlands FML and SML]) Mild Shock <janburse@fastmail.fm> - 2026-10-08 13:12 +0200
Re: Math is the premier AI Victim [Billions of $USD later] (Re: So it was a race against Buzzard. Well done! [Langlands FML and SML]) Gay Banana <ivcoc@sdrde.ru> - 2026-10-08 15:13 +0000
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Superintelligenz ist auch F.U.D. (OAI:The-irrationality-exponent-of-pi-is-2-September-24-2026) Mild Shock <janburse@fastmail.fm> - 2026-10-10 00:35 +0200
Re: Nobody understands relativity "Paul B. Andersen" <paul.b.andersen@paulba.no> - 2026-06-09 21:05 +0200
Re: Nobody understands relativity Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-10 08:39 -0700
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Re: Nobody understands relativity nospam@de-ster.demon.nl (J. J. Lodder) - 2026-06-07 10:54 +0200
Re: Nobody understands relativity marika <marika5000@gmail.com> - 2026-07-02 00:09 +0000
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