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| From | Mild Shock <janburse@fastmail.fm> |
|---|---|
| Newsgroups | sci.logic, comp.lang.prolog, sci.math |
| Subject | Time got dilated by Gemini & Deepseek [The new Shalosh B. Ekhad] (Was: Superintelligenz ist auch F.U.D. ) |
| Date | 2026-10-10 13:00 +0200 |
| Message-ID | <11ad5si$189a6$1@solani.org> (permalink) |
| References | (7 earlier) <11478b2$fv28$2@solani.org> <119omnu$pkou$2@solani.org> <11ab8sg$16ucq$1@solani.org> <11abbds$170ap$1@solani.org> <11abq5p$17a99$2@solani.org> |
Cross-posted to 3 groups.
Hi,
Gemini & Deepseek trying the best to be good
buddies in my Math development now. Still trying
to harden my results. Now computing sigma and tau,
via this little gadget:
% pi(+Integer, +Rational, -Rational)
pi(0, P/Q, R/Q) :- !, R is P-3*Q.
pi(N, P/Q, S) :-
H is 6*P+(2*N-1)^2*Q,
M is N-1,
pi(M, H/P, S).
% up(+Integer, -Float, -Float, -Float)
up(N, U, S, T) :-
pi(N, 6/1, P/Q),
S is msb(Q)/N,
T is msb(abs(Q*31415926535897932384626433832795-
P*10000000000000000000000000000000)
// 10000000000000000000000000000000)/N,
U is 1+S/T.
But the thing seems to be super exponential:
?- up(1000,U,S,T).
U = 2.00336417157275, S = 9.544, T = 9.512.
?- up(10000,U,S,T).
U = 2.0003270111183777, S = 12.8478, T = 12.8436.
?- up(100000,U,S,T).
U = 2.0000321647428647, S = 16.16729, T = 16.16677.
What now? Ok, lets try gcd/2:
% up2(+Integer, -Float, -Float, -Float)
up2(N, U, S, T) :-
pi(N, 6/1, A/B),
C is gcd(A,B),
P is A//C,
Q is B//C,
S is msb(Q)/N,
T is msb(abs(Q*31415926535897932384626433832795-
P*10000000000000000000000000000000)
// 10000000000000000000000000000000)/N,
U is 1+S/T.
Now it becomes exponential:
?- up2(1000,U,S,T).
U = 2.0112874779541445, S = 2.867, T = 2.835.
?- up2(10000,U,S,T).
U = 2.001460158531498, S = 2.8806, T = 2.8764.
?- up2(100000,U,S,T).
U = 2.0001767709152922, S = 2.8856, T = 2.88509.
LoL
Bye
P.S.: Trying to find an exact formula for the GCD
now, or something smaller than the GCD by the help of
Gemini & Deepseek . Despite like Deepseek making
funny errors like gcd(a+b)=gcd(a)+gcd(b), and
Gemini being rather lazy sometimes, I learnt
already a new word Casoratian:
https://mathworld.wolfram.com/Casoratian.html
So Gemini & Deepseek give me a total different
Shalosh B. Ekhad. Not the usual Maple grind, or
Wolfram Alpha grind, more breadth and depth,
when it comes to drawing connections:
Welcome to the Home Page of Doron Zeilberger's servant
Last Update: Aug. 9, 2005
https://sites.math.rutgers.edu/~zeilberg/ekhad.html
Mild Shock schrieb:
> 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
>>>
>>> Mild Shock schrieb:
>>>> Hi,
>>>>
>>>> OpenAI has announced enhancing their Math Swarms
>>>> by an interpretability accelerator prompt recently
>>>> discovered in a Terrence Tao workshop.
>>>>
>>>> So they are now takling:
>>>>
>>>> Mathematician W. Hugh Woodin Explains Continuum Hypothesis
>>>> https://www.youtube.com/watch?v=ZC7wglkBWMM
>>>>
>>>> Infinite fun with infinite worlds
>>>> https://ethz.ch/en/news-and-events/eth-news/news/2020/08/infinite-fun-with-the-infinite-worlds.html
>>>>
>>>>
>>>> A partial Lean 4 formalization is already in
>>>> the making. But the hurdles are high. Some
>>>> speaker of the effort said:
>>>>
>>>> The computer might find a new tamagochi knottheory
>>>> lemma, which will then allow SpaceX to fly to
>>>> workholes, but the hurdle will be hight that
>>>>
>>>> a single human mathematician mind might
>>>> ingest, understand and interpret it.
>>>>
>>>> Bye
>>>>
>>>>
>>
>
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