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 convereges, 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.
Related pages (outside of this work)