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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. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Correspondingly, the primordial example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. In Mathematics, the Sieve of Eratosthenes is a simple ancient Algorithm for finding all Prime numbers up to a specified integer In Mathematics, the Legendre sieve is the simplest method in modern Sieve theory. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms. In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be. The twentieth century of the Common Era began on

One successful approach is to approximate a specific sifted set of numbers (e. g. the set of prime numbers) by another, simpler set (e. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 g. the set of almost prime numbers), which is typically somewhat larger than the original set, and easier to analyze. In Number theory, a Natural number is called k -almost prime If and only if it has exactly k Prime factors counted with More sophisticated sieves also do not work directly with sets per se, but instead count them according to carefully chosen weight functions on these sets (options for giving some elements of these sets more "weight" than others). A weight function is a mathematical device used when performing a sum integral or average in order to give some elements more of a "weight" than others

Modern sieves include the Brun sieve, the Selberg sieve, and the large sieve. In Mathematics, the large sieve is a method of Analytic number theory. One of the original purposes of sieve theory was to try to prove conjectures in number theory such as the twin prime conjecture. The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes While the original broad aims of sieve theory still are largely unachieved, there has been some partial successes, especially in combination with other number theoretic tools. Highlights include:

  1. Brun's theorem, which asserts that the sum of the reciprocals of the twin primes converges (whereas the sum of the reciprocals of the primes themselves diverge);
  2. Chen's theorem, which shows that there are infinitely many primes p such that p + 2 is either a prime or a semiprime (the product of two primes); a closely related theorem of Chen Jingrun asserts that every sufficiently large even number is the sum of a prime and another number which is either a prime or a semiprime. In Mathematics, Brun's theorem is a result in Number theory proved by Viggo Brun in 1919 Chen's theorem was first stated by Chinese Mathematician Chen Jingrun in 1966 with further details of the proof in 1973 In Mathematics, a semiprime (also called biprime or 2- Almost prime, or pq number) is a Natural number that is the product Chen Jingrun ( May 22 1933 – March 19 1996) was a Chinese Mathematician who made significant contributions to Number In Mathematics, the phrase sufficiently large is used in contexts such as P is true for sufficiently large x which is actually shorthand These can be considered to be near-misses to the twin prime conjecture and the Goldbach conjecture respectively. The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics.
  3. The fundamental lemma of sieve theory, which (very roughly speaking) asserts that if one is sifting a set of N numbers, then one can accurately estimate the number of elements left in the sieve after Nε iterations provided that ε is sufficiently small (fractions such as 1/10 are quite typical here). This lemma is usually too weak to sieve out primes (which generally require something like N1 / 2 iterations), but can be enough to obtain results regarding almost primes. In Number theory, a Natural number is called k -almost prime If and only if it has exactly k Prime factors counted with
  4. The Bombieri–Friedlander–Iwaniec theorem, which asserts that there are infinitely many primes of the form a2 + b4. In Analytic number theory, an advanced branch of Mathematics, the Bombieri–Friedlander–Iwaniec theorem asserts that there are infinitely many Prime numbers

The techniques of sieve theory can be quite powerful, but they seem to be limited by an obstacle known as the parity problem, which roughly speaking asserts that sieve theory methods have extreme difficulty distinguishing between numbers with an odd number of prime factors, and numbers with an even number of prime factors. This parity problem is still not very well understood.

Compared with other methods in number theory, sieve theory is comparatively elementary, in the sense that it does not necessarily require sophisticated concepts from either algebraic number theory or analytic number theory. In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers In Mathematics, analytic number theory is a branch of Number theory that uses methods from Mathematical analysis to solve number-theoretical problems Nevertheless, the more advanced sieves can still get very intricate and delicate (especially when combined with other deep techniques in number theory), and entire textbooks have been devoted to this single subfield of number theory; a classic reference is (Halberstam & Richert 1974).

Sieve theory is somewhat related to sieve algorithms such as the general number field sieve, used for factoring large numbers, although sieve theory and sieve algorithms serve different purposes. In Mathematics, the general number field sieve (GNFS is the most efficient classical Algorithm known for factoring integers larger than 100

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