# 布朗常数

1919年，挪威数学家維果·布朗Viggo Brun英语Viggo Brun）证明了所有孪生素数倒数之和收敛于一个数学常数，称为布朗常数(Brun's constant)，记为B2OEIS中的数列A065421）：

${\displaystyle B_{2}=\left({\frac {1}{3}}+{\frac {1}{5}}\right)+\left({\frac {1}{5}}+{\frac {1}{7}}\right)+\left({\frac {1}{11}}+{\frac {1}{13}}\right)+\left({\frac {1}{17}}+{\frac {1}{19}}\right)+\left({\frac {1}{29}}+{\frac {1}{31}}\right)+\cdots }$

Thomas R. Nicely把孪生素数算到1014，估计布朗常数大约为1.902160578。目前最精确的估计是Pascal Sebah和Patrick Demichel在2002年发现的，他们把孪生素数算到了1016

B2 ≈ 1.902160583104.

${\displaystyle B_{4}=\left({\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}\right)+\left({\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+{\frac {1}{19}}\right)+\left({\frac {1}{101}}+{\frac {1}{103}}+{\frac {1}{107}}+{\frac {1}{109}}\right)+\cdots }$

B4 =0.87058 83800 ± 0.00000 00005。

## 参考文献

• Finch, S. R. "Brun's Constant." §2.14 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 133-135, 2003.
• Segal, B. "Généralisation du théorème de Brun." Dokl. Akad. Nauk SSSR, 501-507, 1930.