Brun's theorem

In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B2 (sequence A065421 in the OEIS).

If it were the case that the sum diverged, then that fact would imply that there are infinitely many twin primes.

Brun's constant could be an irrational number only if there are infinitely many twin primes.

Thomas Nicely remarks that after summing the first billion (109) terms, the relative error is still more than 5%.

[1] By calculating the twin primes up to 1014 (and discovering the Pentium FDIV bug along the way), Nicely heuristically estimated Brun's constant to be 1.902160578.

In 2002, Pascal Sebah and Patrick Demichel used all twin primes up to 1016 to give the estimate[2] that B2 ≈ 1.902160583104.

Dominic Klyve showed conditionally (in an unpublished thesis) that B2 < 2.1754 (assuming the extended Riemann hypothesis).

Jie Wu proved that for sufficiently large x, The digits of Brun's constant were used in a bid of $1,902,160,540 in the Nortel patent auction.

[4] Furthermore, academic research on the constant ultimately resulted in the Pentium FDIV bug becoming a notable public relations fiasco for Intel.