Concept
Brun sieve
Related concepts (4)
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the prototypical example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms.
Brun's theorem
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B2 . Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods. The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let denote the number of primes p ≤ x for which p + 2 is also prime (i.
Viggo Brun
Viggo Brun (13 October 1885 – 15 August 1978) was a Norwegian professor, mathematician and number theorist. In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the Brun sieve, which addresses additive problems such as Goldbach's conjecture and the twin prime conjecture. He used it to prove that there exist infinitely many integers n such that n and n+2 have at most nine prime factors, and that all large even integers are the sum of two numbers with at most nine prime factors.
Twin prime
A 'twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin' or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger.