conjecture. Using Vinogradov's method, Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann showed that almost all even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers up to some which can be so written tends towards 1 as increases). In 1930, Lev Schnirelmann proved that any natural number greater than 1 can be written as the sum of not more than prime numbers, where is an effectively computable constant; see Schnirelmann density. Schnirelmann's constant is the lowest number with this property. Schnirelmann himself obtained < . This result was subsequently enhanced by many authors, such as Olivier Ramaré, who in 1995 showed that every even number is in fact the sum of at most 6 primes. The best known result currently stems from the proof of the weak Goldbach conjecture by Harald Helfgott, which directly implies that every even number is the sum of at most 4 primes. In 1924, Hardy and Littlewood showed under the assumption of the generalized Riemann hypothesis that the number of even numbers up to violating the Goldbach conjecture is much less than for small . In 1948, using sieve theory, Alfréd Rényi showed that every sufficiently large even number can be written as the sum of a prime and an almost prime with at most K factors. Chen Jingrun showed in 1973 using the methods of sieve theory that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). See Chen's theorem for further information. In 1975, Hugh Montgomery and Robert Charles Vaughan showed that "most" even numbers are expressible as the sum of two primes. More precisely, they showed that there exist positive constants and such that for all sufficiently large numbers , every even number less than is the sum of two primes, with at most exceptions. In particular, the set of even integers that are not the sum of two primes has density zero. In 1951, Yuri Linnik proved the existence of a constant such that every sufficiently large even number is the sum of two primes and at most powers of 2. Roger Heath-Brown and Jan-Christoph Schlage-Puchta found in 2002 that works. Related problems Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at