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.

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