# Brun's constant

In 1919 Brun showed that the sum of the reciprocals of the twin primes converges to a sum now called **Brun's Constant**:

(1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + ...

Had this series diverged, then we would have a proof of the twin primes conjecture. But since it converges, we do not yet know if there are infinitely many twin primes.

By calculating the twin primes up to 10^{14} (and discovering the infamous Pentium bug along the way),
Nicely heuristically estimated Brun's constant to be 1.902160578. More recently he has improved this estimate
to 1.9021605824 by using the twins to 1.6^{.}10^{15}.

**See Also:** TwinPrime, TwinPrimeConstant

**Related pages** (outside of this work)

- The sum of the reciprocals of all primes diverges
- Twin primes and the Pentium bug
- Thomas Nicely's homepage with links to some of his work

**References:**

- Nicely95
T. Nicely, "Enumeration to 10^{14}of the twin primes and Brun's constant,"Virginia Journal of Science,46:3 (1995) 195--204.MR 97e:11014(Abstract available) [Available at http://www.trnicely.net/index.html]

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