Brun's constant

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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 in OEIS):

[B_2 = \left(\frac + \frac\right)+ \left(\frac + \frac\right)+ \left(\frac + \frac\right)+ \left(\frac + \frac\right)+ \left(\frac + \frac\right) + \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 of its rationality 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 and Patrick Demichel in 2002, using all twin primes up to 10¹⁶:

B₂ ≈ 1.902160583104

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:

[B_4 = \left(\frac + \frac + \frac + \frac\right)+ \left(\frac + \frac + \frac + \frac\right)+ \left(\frac + \frac + \frac + \frac\right) + \cdots]

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.

See also

• twin prime
• twin prime constant
• twin prime conjecture
• Meissel-Mertens constant

External links

• [Nicely's article on twins enumeration and Brun's constant]
• [Computation of Brun's constant]
• [Eric W. Weisstein. "Brun's Constant." From MathWorld--A Wolfram Web Resource.]

 

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