Brun's constant
The Brun's constant is a mathematical constant in the field of number theory. It is named after the mathematician Viggo Brun.
Brun's constant for twin primes
In 1919 the mathematician Viggo Brun showed that the sum of the reciprocals of all twin primes (pairs of primes whose difference is 2 ) converges. The limit of this sum is called Brun's constant for twin primes:
This result of analytic number theory is surprising at first glance, since the sum of the reciprocals of all the primes diverges, as evidenced in the 18th century by Leonhard Euler. Would also divergent, there would be a proof of the conjecture to open today, that there are infinitely many twin primes (Alphonse de Polignac ( 1817-1890 ), 1849). From the convergence, however, does not suggest the opposite.
Calculation
The idea to calculate is that the summation is performed first as far as possible and then the missing residual is estimated. So have Daniel Shanks and John William Wrench, Jr. (1911-2009) used all twin primes below 2106.
An estimate of
Comes from Pascal Sebah from 2002, all the twin primes looked to 1016 for this purpose. The calculation of, however, is extremely difficult, on the one hand, since the series converges very slowly, on the other hand, since finding all large twin primes is extremely complicated (see also: primality testing ).
The most precise estimate is (as of March 16, 2010 )
For this, the reciprocals of all twin primes 19,831,847,025,792 were summed below 2.1016:
And the remainder term is estimated.
Brun's constant for prime quadruplets
In addition, there is the Brun's constant for prime quadruplets, pairs of twin primes, which have a distance of 4 ( this is the smallest possible distance between two twin primes to each other). The first three prime quadruplets (5, 7, 11, 13), ( 11, 13, 17, 19) and ( 101, 103, 107, 109), so
Since all summands occur from in and up to and no summands are duplicated also converges this series. It has the value (as of March 16, 2010 )