In this chapter, we shall prove one of the most famous results in additive prime number theory: Chen’s theorem that every sufficiently large even integer can be written as the sum of an odd prime and a number that is either prime or the product of two primes.
KeywordsPrime Number Dirichlet Character Implied Constant Prime Number Theorem Large Sieve
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- Chen [10, 11] announced his theorem in 1966 but did not publish the proof until 1973, apparently because of difficulties arising from the Cultural Revolution in China. An account of Chen’s original proof appears in Halberstam and Richert’s Sieve Methods . The proof in this chapter is based on unpublished notes and lectures of Henryk Iwaniec . The argument uses standard results from multiplicative number theory (Dirichlet characters, the large sieve, and the Siegel-Walfisz and Bombieri-Vinogradov theorems), all of which can be found in Davenport . Other good references for these results are the monographs of Montgomery  and Bombieri . For bilinear form inequalities, see Bombieri, Friedlander, and Iwaniec .Google Scholar