Brun's theorem

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In mathematics, Brun's theorem is a result in number theory proved by Viggo Brun in 1919. It has historical importance in the introduction of sieve methods.

Let P(x) denote the number of primes p ≤ x for which p + 2 is also prime (i.e. P(x) is the number of twin primes). Then, for x ≥ 3, we have

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

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

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

converges, and its value is known as Brun's constant. Unlike the case for all prime numbers, we cannot conclude from this result that there are an infinite number of twin primes.

[edit] References

Eric W. Weisstein, Brun's Theorem at MathWorld.

Brun, V. "La serie 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., les dénominateurs sont nombres premiers jumeaux est convergente où finie." Bull. Sci. Math. 43, p.124-128, 1919. The original paper