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)