Definition:Brun's Constant

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Definition

Brun's constant is the sum of the series consisting of the reciprocals of the twin primes:

$B_2 := \left({\dfrac 1 3 + \dfrac 1 5}\right) + \left({\dfrac 1 5 + \dfrac 1 7}\right) + \left({\dfrac 1 {11} + \dfrac 1 {13} }\right) + \left({\dfrac 1 {17} + \dfrac 1 {19} }\right) + \left({\dfrac 1 {29} + \dfrac 1 {31} }\right) + \cdots$


Its approxmiate decimal expansion is:

$B_2 \approx 1 \cdotp 90216 \, 05831 \, 04 \ldots$

This sequence is A065421 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Estimates of its value are occasionally refined as further work is done to establish its nature.


Also defined as

Some sources do not include one of the instances of $\dfrac 1 5$.


Also see


Source of Name

This entry was named for Viggo Brun.


Sources