sieve after iterations provided that 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 iterations), but can be enough to obtain results regarding almost primes. The Friedlander–Iwaniec theorem, which asserts that there are infinitely many primes of the form . Zhang's theorem , which shows that there are infinitely many pairs of primes within a bounded distance. The Maynard–Tao theorem generalizes Zhang's theorem to arbitrarily long sequences of primes. 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, 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 and a more modern text is . 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. Literature External links References