Brun's theorem

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 is convergent with a finite value known as Brun's constant. 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

:

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

:$sumlimits\_\{\; p\; ,\; :\; ,\; p\; +\; 2\; in\; mathbb\{P\}\; \}\; \{left(\; \{frac\{1\}\{p\}\; +\; frac\{1\}$p + 2} ight)} = left( {frac{1}{3} + frac{1}{5 ight) + left( {frac{1}{5} + frac{1}{7 ight) + left( {frac{1}11 + frac{1}13} ight) + 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 are an infinite number of twin primes.

ee also

*Proof that the sum of the reciprocals of the primes diverges

References

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* Reprinted Providence, RI: Amer. Math. Soc., 1990.
* Contains a more modern proof.

External links

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* Sebah, Pascal and Xavier Gourdon, [http://numbers.computation.free.fr/Constants/Primes/twin.pdf Introduction to twin primes and Brun's constant computation] , 2002. A modern detailed examination.