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 10¹⁴ (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^.10¹⁵.

See Also: TwinPrime, TwinPrimeConstant

Related pages (outside of this work)

• The sum of the reciprocals of all primes diverges
• Thomas Nicely's homepage with links to some of his work

References:

Nicely95

T. Nicely, "Enumeration to 10¹⁴ 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]