Chen’s theorem with small primes

Abstract

Let N be a sufficiently large even integer. Let p denote a prime and P ₂ denote an almost prime with at most two prime factors. In this paper, it is proved that the equation N = p + P ₂ (p ≤ N ^0.945) is solvable.

Keywords

Chen’s Theorem / Sieve method / Mean value theorem

References

[1] Cai Y. C.. Chen’s theorem with small primes. Acta Math. Sin. (English Series), 2002, 18(3): 597-604

[2] Cai Y. C., Lu M. G.. On Chen’s Theorem, Analytic Number Theory, Beijing/Kyoto, 1999, 2002, Dordrecht: Kluwer Acad. Publ. 99-119

[3] Cai Y. C.. On Chen’s theorem (II). J. Number Theory, 2008, 128(5): 1336-1357

[4] Chen J. R.. On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao, 1966, 17: 385-386

[5] Chen J. R.. On the representation of a large even integer as the sum of a prime and the product of at most two primes. Sci. Sin., 1973, 16: 157-176

[6] Chen J. R.. On the Goldback’s problem and the sieve methods. Sci. Sin., 1978, 21: 701-739

[7] Chen J. R.. On the representation of a large even integer as the sum of a prime and the product of at most two primes (II). Sci. Sin., 1978, 21: 421-430

[8] Chen J. R.. On the representation of a large even integer as the sum of a prime and the product of at most two primes (II) (in Chinese). Sci. Sin., 1978, 21: 477-494

[9] Halberstam H., Richert H. E.. Sieve Methods, 1974, London: Academic Press

[10] Iwaniec H.. Rosser’s Sieve, Recent Progress in Analytic Number Theory II, 1981, London: Academic Press 203-230

[11] Pan C. D., Pan C. B.. Goldbach Conjecture, 1992, Beijing: Science Press 175-176

[12] Pan C. D., Pan C. B.. Goldbach Conjecture (in Chinese), 1981, Beijing: Science Press 239-251

[13] Wu J.. Theoremes generalises de Bombieri-Vinogradov dans les petits applications, intervalles. Quart. J. Math. (Oxford), 1993, 44: 109-128

[14] Wu J.. Chen’s double sieve, Goldbach’s conjecture and the twin prime problem. Acta Arith., 2004, 114: 215-273