An elementary proof or pi irrationality = 2 [Natural Intelligence]

From	Mild Shock <janburse@fastmail.fm>
Newsgroups	sci.logic, comp.lang.prolog, sci.math
Subject	An elementary proof or pi irrationality = 2 [Natural Intelligence]
Date	2026-10-09 19:39 +0200
Message-ID	<11ab8sg$16ucq$1@solani.org> (permalink)
References	(4 earlier) <10sriht$13app$2@solani.org> <10srjsg$13bfs$1@solani.org> <10ssije$13umd$3@solani.org> <11478b2$fv28$2@solani.org> <119omnu$pkou$2@solani.org>

Cross-posted to 3 groups.

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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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