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:
- $B_2 := \paren {\dfrac 1 3 + \dfrac 1 5} + \paren {\dfrac 1 5 + \dfrac 1 7} + \paren {\dfrac 1 {11} + \dfrac 1 {13} } + \paren {\dfrac 1 {17} + \dfrac 1 {19} } + \paren {\dfrac 1 {29} + \dfrac 1 {31} } + \cdots$
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
- 1974: Daniel Shanks and John W. Wrench, Jr.: Brun's Constant (Math. Comp. Vol. 28: pp. 293 – 299) www.jstor.org/stable/2005836
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 90195 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 90216 \, 054 \ldots$
- Weisstein, Eric W. "Brun's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrunsConstant.html