GWB
Posts:
16
Registered:
7/22/08


Brun's constant and Phi relationship?
Posted:
Dec 31, 2012 4:12 PM


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 PH200412b.pdf), in which he proposes a method for distinguishing coincidental data compression from noncoincidental 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 noncoincidental.
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

