Sources
open directory project
Help build the largest human-edited directory on the web.
Submit a site - Become an editor
Wikipedia. Licensed under the GNU Free Documentation License.
Are you an expert in this subject? Join the discussion and share your knowledge at Wikipedia.org.
Brun's theorem
From Wikipedia, the free encyclopedia
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
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
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.
- 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
- Landau, E. Elementare Zahlentheorie. Leipzig, Germany: Hirzel, 1927. Reprinted Providence, RI: Amer. Math. Soc., 1990.
- W.J. LeVeque, Fundamentals of Number Theory, New York, Dover, 1996. Contains a more modern proof.
- Sebah, Pascal and Xavier Gourdon, Introduction to twin primes and Brun's constant computation, 2002. A modern detailed examination.