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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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GWB

Posts: 16
Registered: 7/22/08
Brun's constant and Phi relationship?
Posted: Dec 31, 2012 4:12 PM
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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



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