Brun's theorem

The convergence to Brun's constant.

In mathematics, Brun's theorem is a result in number theory proved by Viggo Brun in 1919. It states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) is convergent with a finite value known as Brun's constant, usually denoted by B₂ (sequence A065421 in the OEIS). It has historical importance in the introduction of sieve methods.

Let $\pi _{2}(x)$ denote the number of primes p ≤ x for which p + 2 is also prime (i.e. $\pi _{2}(x)$ is the number of twin primes with the smaller at most x). Then, for x ≥ 3, we have

$\pi _{2}(x)<c{\frac {x}{(\log x)^{2}}}(\log \log x)^{2}$

for some positive constant c.

This result shows that the sum of the reciprocals of the twin primes converges; in other words the p involved are a small set. In explicit terms the sum

$\sum \limits _{p\,:\,p+2\in \mathbb {P} }{\left({{\frac {1}{p}}+{\frac {1}{p+2}}}\right)}=\left({{\frac {1}{3}}+{\frac {1}{5}}}\right)+\left({{\frac {1}{5}}+{\frac {1}{7}}}\right)+\left({{\frac {1}{11}}+{\frac {1}{13}}}\right)+\cdots$

either has finitely many terms or has infinitely many terms but is convergent: its value is known as Brun's constant. Unlike the case for all prime numbers, we cannot conclude from this result that there is an infinite number of twin primes.

Brun's constant

By calculating the twin primes up to 10¹⁴ (and discovering the Pentium FDIV bug along the way), Thomas R. Nicely heuristically estimated Brun's constant to be 1.902160578. Nicely has extended his computation to 1.6×10¹⁵ 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:

B₂ ≈ 1.902160583104.

It is based on extrapolation from the sum 1.830484424658... for the twin primes below 10¹⁶. Richard Crandall and Carl Pomerance reported that it is known rigorously that 1.83 < B₂ < 2.347. Dominic Klyve showed conditionally that B₂ < 2.1754 (assuming the extended Riemann hypothesis).

The digits of Brun's constant was used in a bid of $1,902,160,540 in the Nortel patent auction. The bid was posted by Google and was one of three Google bids based on mathematical constants.

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:

$B_{4}=\left({\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}\right)+\left({\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+{\frac {1}{19}}\right)+\left({\frac {1}{101}}+{\frac {1}{103}}+{\frac {1}{107}}+{\frac {1}{109}}\right)+\cdots$

with value:

B₄ = 0.87058 83800 ± 0.00000 00005, the error range having a 99% confidence level according to Nicely.

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. Wolf derived an estimate for the Brun-type sums B_n of 4/n.

Further results

Let $C_{2}=0.6601\ldots$ (sequence A005597 in the OEIS). Then it is conjectured that

$\pi _{2}(x)\sim 2C_{2}{\frac {x}{(\log x)^{2}}}.$

In particular,

$\pi _{2}(x)<(2C_{2}+\varepsilon ){\frac {x}{(\log x)^{2}}}$

for every $\varepsilon >0$ and all sufficiently large x.

Many special cases of the above have been proved. Most recently, Jie Wu proved that for sufficiently large x,

$\pi _{2}(x)<4.5{\frac {x}{(\log x)^{2}}}$

where 4.5 corresponds to $\varepsilon \approx 3.18$ in the above.

See also

• Proof that the sum of the reciprocals of the primes diverges
• Meissel–Mertens constant

References

• Viggo Brun (1915). "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare". Archiv for Math. Og Naturvid. B34 (8).
• Template:Fr Viggo Brun (1919). "La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., où les dénominateurs sont nombres premiers jumeaux est convergente ou finie". Bulletin des sciences mathématiques. 43: 100–104, 124–128.
• Alina Carmen Cojocaru (2005). An introduction to sieve methods and their applications. London Mathematical Society Student Texts. Vol. 66. Cambridge University Press. pp. 73–74. ISBN 0-521-61275-6. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• E. Landau (1927). Elementare Zahlentheorie. Leipzig, Germany: Hirzel. Reprinted Providence, RI: Amer. Math. Soc., 1990.
• William Judson LeVeque (1996). Fundamentals of Number Theory. New York: Dover. pp. 1–288. ISBN 0486689069. Contains a more modern proof.
• J. Wu (2004). "Chen's double sieve, Goldbach's conjecture and the twin prime problem". Acta Arithmetica. 114 (3): pp. 215–273. arXiv:0709.3764. doi:10.4064/aa114-3-2. {{cite journal}}: |pages= has extra text (help)

External links

• Weisstein, Eric W. "Brun's Theorem". MathWorld.
• Brun's constant at PlanetMath.
• Sebah, Pascal and Xavier Gourdon, Introduction to twin primes and Brun's constant computation, 2002. A modern detailed examination.
• Wolf's article on Brun-type sums