- Brun's constant
In 1919
Viggo Brun showed that the sum of the reciprocals of thetwin prime s (pairs ofprime number s which differ by 2) converges to amathematical constant now called Brun'sconstant for twin primes and usually denoted by "B"2 OEIS|id=A065421::B_2 = left(frac{1}{3} + frac{1}{5} ight)+ left(frac{1}{5} + frac{1}{7} ight)+ left(frac{1}{11} + frac{1}{13} ight)+ left(frac{1}{17} + frac{1}{19} ight)+ left(frac{1}{29} + frac{1}{31} ight) + cdots
in stark contrast to the fact that the sum of the reciprocals of all primes is divergent. Had this series diverged, we would have a proof of the
twin prime conjecture . But since it converges, we do not yet know if there are infinitely many twin primes. Similarly, if it were ever to be proved that Brun's constant was irrational, the twin primes conjecture would follow immediately, whereas a proof that it is rational wouldn't decide it either way.Brun's sieve was refined by
J.B. Rosser , G. Ricci and others.By calculating the twin primes up to 1014 (and discovering the infamous
Pentium FDIV bug along the way), Thomas R. Nicely heuristically estimated Brun's constant to be 1.902160578. The best estimate to date was given by Pascal Sebah andPatrick Demichel in 2002, using all twin primes up to 1016:: "B"2 ≈ 1.902160583104.
While 1.9 < "B"2 is shown, no real number N is known such that "B"2 < N.
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 smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by "B"4, is the sum of the reciprocals of all prime quadruplets::B_4 = left(frac{1}{5} + frac{1}{7} + frac{1}{11} + frac{1}{13} ight)+ left(frac{1}{11} + frac{1}{13} + frac{1}{17} + frac{1}{19} ight)+ left(frac{1}{101} + frac{1}{103} + frac{1}{107} + frac{1}{109} ight) + cdots
with value:
:"B"4 = 0.87058 83800 ± 0.00000 00005.
This constant should not be confused with the Brun's constant for
cousin prime s, prime pairs of the form ("p", "p" + 4), which is also written as "B"4. Wolf derived an estimate for the Brun-type sums "B"n of 4/"n". This gives the estimate for "B"n of 2, about 5% higher than the true value.ee also
*
Twin prime conjecture
*Meissel-Mertens constant References
*
*
*External links
* [http://www.trnicely.net/twins/twins2.html Nicely's article on twins enumeration and Brun's constant]
* [http://numbers.computation.free.fr/Constants/Primes/twin.html Computation of Brun's constant]
*
*
* [http://www.ift.uni.wroc.pl/~mwolf/ Wolf's article on Brun-type sums]
Wikimedia Foundation. 2010.