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In 1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a mathematical constant now called Brun's constant for twin primes and usually denoted by B2 (sequence A065421 in OEIS): Viggo Brun (October 13, 1882 - August 15, 1978) was a Norwegian mathematician. ...
A twin prime is a prime number that differs from another prime number by two. ...
In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...
A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...
The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...
 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. In the third century BC, Euclid proved the existence of infinitely many prime numbers. ...
The twin prime conjecture is a famous problem in number theory that involves prime numbers. ...
In mathematics, an irrational number is any real number that is not a rational number, i. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
Brun's sieve was refined by J.B. Rosser, G. Ricci and others. John Barkley Rosser Sr. ...
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 and Patrick Demichel in 2002, using all twin primes up to 1016: On October 30, 1994, Professor Thomas Nicely who was then at Lynchburg College reported a bug in the Pentium floating point unit. ...
The Pentium FDIV bug was a bug in Intels original Pentium floating point unit. ...
For album titles with the same name, see 2002 (album). ...
- B2 ≈ 1.902160583104
While 1.9 < B2 is shown, no real number N is known such that B2 < 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 B4, is the sum of the reciprocals of all prime quadruplets: A prime quadruplet is a group of four primes, consisting of two pairs of twin primes separated only by three non-primes, specifically, a multiple of 2, a multiple of 15 and another multiple of 2. ...
 with value: - B4 = 0.87058 83800 ± 0.00000 00005.
This constant should not be confused with the Brun's constant for cousin primes, prime pairs of the form (p, p + 4), which is also written as B4. In mathematics, a cousin prime is a pair of prime numbers that differ by four; compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six. ...
See also
A twin prime is a prime number that differs from another prime number by two. ...
The twin prime conjecture is a famous problem in number theory that involves prime numbers. ...
The Meissel-Mertens constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm: Its value is approximately M ≈ 0. ...
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