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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.
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Sieves in Number Theory -

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. Furthermore, in some modern applications, sieves are used not to estimate the size of a sifted set, but to produce a function that is large on the set and mostly small outside it, while being easier to analyze than the characteristic function of the set.

One of the original purposes of sieve theory was to try to prove conjectures in number theory such as the twin prime conjecture. While the original broad aims of sieve theory still are largely unachieved, there have been some partial successes, especially in combination with other number theoretic tools.

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Highlights include:. 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. The book of H. Halberstam and H.

Richert had actually been conceived in the mid's. The initial stimulus had been provided by the paper of W. Jurkat and Richert, which determined the sifting limit for the linear sieve, using a combination of the ,A2 method of A.

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Selberg with combinatorial ideas which were in themselves of great importance and in terest. One of the declared objectives in writing their book was to place on record the sharpest form of what they called Selberg sieve theory available at the time.

Sieve of Eratosthenes

At the same time combinatorial methods were not neglected, and Halber stam and Richert included an account of the purely combinatorial method of Brun, which they believed to be worthy of further examination. So the prime numbers are the unmarked ones: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, Thanks to Krishan Kumar for providing above explanation.

Implementation: Following is the implementation of the above algorithm. In the following implementation, a boolean array arr[] of size n is used to mark multiples of prime numbers. You may also like to see : Segmented Sieve. Sieve of Eratosthenes in 0 n time complexity.

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