Let $N$ denote a sufficiently large even integer. In this paper itis proved that for $0.941 \leq \theta \leq 1$, the equation$$N=p+P_2,\hspace{10mm} p\leq N^{\theta}$$is solvable, where $p$ is a prime and $P_2$ is an almost prime withat most two prime factors. The range $0.941 \leq \theta \leq 1$ extended the previous one $0.945 \leq \theta \leq 1$.
References
E. Bombieri, On the large sieve, Mathematika, 12 (1965), 201-225. MR197425 0136.33004 10.1112/S0025579300005313E. Bombieri, On the large sieve, Mathematika, 12 (1965), 201-225. MR197425 0136.33004 10.1112/S0025579300005313
Y. C. Cai, On Chen's theorem II, Journal of Number Theory, 128(5) (2008), 1336-1357. 1146.11052 10.1016/j.jnt.2007.04.010Y. C. Cai, On Chen's theorem II, Journal of Number Theory, 128(5) (2008), 1336-1357. 1146.11052 10.1016/j.jnt.2007.04.010
Y. C. Cai, Chen's theorem with small primes, Acta Math. Sinnica, 18 (2002), 597-604. MR1929234 1044.11089 10.1007/s101140200168Y. C. Cai, Chen's theorem with small primes, Acta Math. Sinnica, 18 (2002), 597-604. MR1929234 1044.11089 10.1007/s101140200168
J. R. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Kexue Tongbao, 17 (1966), 385-386. J. R. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Kexue Tongbao, 17 (1966), 385-386.
J. R. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sin., 16 (1973), 157-176. MR434997 0319.10056J. R. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sin., 16 (1973), 157-176. MR434997 0319.10056
J. R. Chen, On the Goldbach's problem and the sieve methods, Sci. Sin., 21 (1978), 701-739. 0399.10046J. R. Chen, On the Goldbach's problem and the sieve methods, Sci. Sin., 21 (1978), 701-739. 0399.10046
H. Halberstam, D. R. Heath-Brown and H. E. Richert, Almost-primes in short intervals, Recent progress in analytic number theory I, 69-101, Academic Press, 1981. H. Halberstam, D. R. Heath-Brown and H. E. Richert, Almost-primes in short intervals, Recent progress in analytic number theory I, 69-101, Academic Press, 1981.
G. Harman, Prime-detecting sieves, London Mathematical Society Monographs Series, 33, Princeton University Press, Princeton, NJ, 2007. 1220.11118G. Harman, Prime-detecting sieves, London Mathematical Society Monographs Series, 33, Princeton University Press, Princeton, NJ, 2007. 1220.11118
D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity, Can. J. Math., XXXIV(6) (1982), 1365-1377. MR678676 0478.10024 10.4153/CJM-1982-095-9D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity, Can. J. Math., XXXIV(6) (1982), 1365-1377. MR678676 0478.10024 10.4153/CJM-1982-095-9
C. H. Jia, Almost all short intervals containing prime numbers, Acta Arith., LXXVI(1) (1996), 21-84. MR1390568 0841.11043 10.4064/aa-76-1-21-84C. H. Jia, Almost all short intervals containing prime numbers, Acta Arith., LXXVI(1) (1996), 21-84. MR1390568 0841.11043 10.4064/aa-76-1-21-84
C. H. Jia, On the exceptional set of Goldbach numbers in a short interval, Acta Arith., 77.3 (1996), 207-287. MR1410337 0863.11066 10.4064/aa-77-3-207-287C. H. Jia, On the exceptional set of Goldbach numbers in a short interval, Acta Arith., 77.3 (1996), 207-287. MR1410337 0863.11066 10.4064/aa-77-3-207-287
H. L. Montgomery, Topics in Multiplicative Number Theory, Springer-Verlag, Berlin, 1971. 0216.03501H. L. Montgomery, Topics in Multiplicative Number Theory, Springer-Verlag, Berlin, 1971. 0216.03501
E. C. Titchmarsh, The theory of the Riemann Zeta-function, Clarendon Press, Oxford, 1951. 0042.07901E. C. Titchmarsh, The theory of the Riemann Zeta-function, Clarendon Press, Oxford, 1951. 0042.07901
Y. J. Li and Y. C. Cai, Chen's theorem with small primes, Chin. Ann. Math. Ser. B, 32(3) (2011), 387-396. 1297.11119 10.1007/s11401-011-0645-4Y. J. Li and Y. C. Cai, Chen's theorem with small primes, Chin. Ann. Math. Ser. B, 32(3) (2011), 387-396. 1297.11119 10.1007/s11401-011-0645-4
J. Wu, Theoremes generalises de Bombieri-Vinogradov dans les petits applications, intervalles, Quart. J. Math., $($Oxford$)$, 44 (1993), 109-128. 0801.11038 10.1093/qmath/44.1.109J. Wu, Theoremes generalises de Bombieri-Vinogradov dans les petits applications, intervalles, Quart. J. Math., $($Oxford$)$, 44 (1993), 109-128. 0801.11038 10.1093/qmath/44.1.109
J. Wu, Sur la suite des nombres premiers jumeaux, Acta Arith, LV (1990), 365-394. 0662.10033 10.4064/aa-55-4-365-394J. Wu, Sur la suite des nombres premiers jumeaux, Acta Arith, LV (1990), 365-394. 0662.10033 10.4064/aa-55-4-365-394
