The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let denote the number of primes p ≤ x for which p + 2 is also prime. Then, for x ≥ 3, we have That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor. It follows from this bound that the sum of the reciprocals of the twin primes converges, or stated in other words, the twin primes form a small set. In explicit terms the sum either has finitely many terms or has infinitely many terms but is convergent: its value is known as Brun's constant. The fact that the sum of the reciprocals of the prime numbers diverges implies that there are infinitely many prime numbers. Because the sum of the reciprocals of the twin primes instead converges, it is not possible to conclude from this result that there are finitely many or infinitely many twin primes. Brun's constant could be an irrational number only if there are infinitely many twin primes.
Numerical estimates
The series converges extremely slowly. Thomas Nicely remarks that after summing the first one billion terms, the relative error is still more than 5%.. By calculating the twin primes up to 1014, Nicely heuristically estimated Brun's constant to be 1.902160578. Nicely has extended his computation to 1.6 as of 18 January 2010 but this is not the largest computation of its type. In 2002, Pascal Sebah and Patrick Demichel used all twin primes up to 1016 to give the estimate that B2 ≈ 1.902160583104. Hence,
Year
B2
# of twin primes used
by
1976
1.902160540
1
Brent
1996
1.902160578
1
Nicely
2002
1.902160583104
1
Sebah and Demichel
The last is based on extrapolation from the sum 1.830484424658... for the twin primes below 1016. Dominic Klyve showed conditionally that B2 < 2.1754. It has been shown unconditionally that B2 < 2.347. There is also a Brun's constant for prime quadruplets. A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4. The first prime quadruplets are,,. Brun's constant for prime quadruplets, denoted by B4, is the sum of the reciprocals of all prime quadruplets: with value: This constant should not be confused with the Brun's constant for cousin primes, as prime pairsof the form, which is also written as B4. Wolf derived an estimate for the Brun-type sums Bn of 4/n.
Further results
Let be the twin prime constant. Then it is conjectured that In particular, for every and all sufficiently largex. Many special cases of the above have been proved. Most recently, Jie Wu proved that for sufficiently large x, where 4.5 corresponds to in the above.
The digits of Brun's constant were used in a bid of $1,902,160,540 in the Nortel patent auction. The bid was posted by Google and was one of three Google bids based on mathematical constants.
