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Sieve theory
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== Techniques of sieve theory == The techniques of sieve theory can be quite powerful, but they seem to be limited by an obstacle known as the ''[[parity problem (sieve theory)|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]]. 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 {{harv|Halberstam|Richert|1974}} and a more modern text is {{harv|Iwaniec|Friedlander|2010}}. The sieve methods discussed in this article are not closely related to the [[integer factorization]] sieve methods such as the [[quadratic sieve]] and the [[general number field sieve]]. Those factorization methods use the idea of the [[sieve of Eratosthenes]] to determine efficiently which members of a list of numbers can be completely factored into small primes.
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