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Concept
Brun's theorem
Summary
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 . Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods.
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 (i.e. is the number of twin primes with the smaller at most x). Then, 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.
If it were the case that the sum diverged, then that fact would imply that there are infinitely many twin primes. 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.
The series converges extremely slowly. Thomas Nicely remarks that after summing the first billion (109) terms, the relative error is still more than 5%.
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. 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,
The last is based on extrapolation from the sum 1.830484424658... for the twin primes below 1016.
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