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Topic: Brun's constant and Phi relationship?
Replies: 15   Last Post: Jul 7, 2013 8:08 AM

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dan73

Posts: 465
From: ct
Registered: 6/14/08
Re: Brun's constant and Phi relationship?
Posted: Jul 7, 2013 8:08 AM
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> About two weeks ago I observed an interesting
> relationship between
> Brun's constant (B = 1.902160583) and the Golden
> Ratio: 1.902160583^2/
> (Phi + 2) = 1.00004999808. I've presented it to Jim
> Markovitch, the
> author of the paper "Coincidence, data compression,
> and Mach?s concept
> of "economy of thought"
> (http://cogprints.org/3667/1/APRI-
> PH-2004-12b.pdf), in which he proposes a method for
> distinguishing
> coincidental data compression from non-coincidental
> ones. Since no
> more than 10 or 11 digits of Brun's constant are
> currently known, no
> analysis whether this is a coincidence or not has
> been made. However
> Jim has found a few more interesting relationships
> between B and Phi,
> the most remarkable of which being
> (1.902160583^2/Phi)^2 =
> 5.00049999334.
>
> These allow for the following nice approximations:
>
> B ~ sqrt(1.00005*(Phi + 2)) and
>
> B ~ sqrt(Phi*sqrt(5.0005))
>
> which approximate B to 9 and 10 significant,
> respectively.
>
> Here is how they fare to a couple of recent
> computations of B:
>
> 1.902160583209 ± 0.000000000781 Thomas R. Nicely,
> March 2010 (http://
> www.trnicely.net/counts.html)
>
> 1.902160583104 Pascal Sebah, 2002
> (http://
> numbers.computation.free.fr/Constants/Primes/twin.pdf)
>
> 1.902160583633
> sqrt(Phi*sqrt(5.0005))
>
> 1.902160584822 sqrt(1.00005*(Phi +
> 2))
>
> Please notice I don't claim this is non-coincidental.
>
> Here is a link to an article by Ivars Peterson on
> Prime Twins and
> Brun's constant:
>
> http://www.maa.org/mathland/mathtrek_6_4_01.html
>
> If B is indeed an irrational number then there will
> surely be
> infinitely many twin primes.
>
> G. W. Barbosa


There are many coincidences in mathematics.
One of my own -- sqrt(pi+1)*pi^2 ~ e^3
My favorite is Ramanujan's formula --
e^(pi(sqrt(163)) =262537412640768743.99999999999925...
Many many more --- see
Wikipedia -- Mathematical coincidences and
also Almost integer.

Cheers,

Dan



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