> 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.