Almost twin primes and Chen's theorem 

Page maintained by Jean-Claude Evard. Last update: June 23, 2003.

AMS classification numbers  Link :  11P32  Link , 11N36  Link .

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 Definitions 

1. We say that an integer greater than 1 is an r-almost prime if and only if it is 
     the product of at most r primes.
2. The set of all r-almost primes is denoted by Pr .
3. We say that a positive integer is an almost prime if and only if it is a 
     2-almost prime.
4. We say that a pair of positive integers are r-almost twin primes if and only if 
     one integer of the pair is a prime and the other is an r-almost prime.
5. We say that a pair of positive integers are almost twin primes if and only if 
     they are 2-almost twin primes .

 Historical result about almost twin primes 

Chen's theorem says that both the Goldbach's conjecture 
and the twin prime conjecture are almost true, in the following sense:

1. There exist an integer N such that every even integer greater than N is the
    sum of a prime and an almost prime.
2. There exist infinitely many almost twin primes.

     The statement of Chen's theorem was published in 1966 and its proof was  
published in 1973 by Jing Run Chen (See references in the list below).

Web pages about almost twin primes

1. Web page on Chen's theorem  Link  maintained by Eric Weisstein  Link   
on the Web site MathWorld  Link  at Wolfram Research  Link .

 Papers related to Chen's theorem 

 Published in 2002 
Yingchun Cai and Minggao Lu
On Chen's theorem
Analytic number theory
(Beijing/Kyoto, 1999), 99--119, 
Dev. Math., 6, 
Kluwer Acad. Publ.,Dordrecht, 2002.
Reviewed by G. Greaves  Link .
 The paper contains the most recent improvement of the original Chen's theorem 
Mathematical Review number: 2003a:11130
AMS classification numbers: 11P32 (11N36)

 

 

 Published in 1999 
Ying Chun Cai and Ming Gao Lu, Chen's theorem in short intervals, Acta Arith. 
 91 (1999), no. 4, 311--323.  This paper contains improvements of results 
 published by Jie Wu in 1994 and by Saverio Salerno and Antonio Vitolo in 
 1993. The authors published a draft of this paper in 1998. The spelling of their 
 first names has been slightly changed.  Review by Giovanni Coppola  Link .

 J. B. Friedlander and D. A. Goldston, Note on a variance in the distribution 
of primes, Number theory in progress, Vol. 2, 841--848, de Gruyter, 1999. 
Review by Cem Y. Yildirim  Link . 

D. I. Tolev, Arithmetic progressions of prime-almost-prime twins, Acta 
Arith. 88 (1999), no. 1, 67--98. Review by Don Redmond  Link .

 Published in 1998 
Minggao Lu and Yingchun Cai, Chen's theorem in short intervals, Chinese Sci. 
Bull. 43 (1998), no. 16, 1401--1403.  This paper seems to be a draft of the   
 paper published by the same authors in 1999. The spelling of their first names 
 has been slightly changed.  Review by Alessandro Languasco  Link .

 Published in 1997 
Koichi Kawada, Note on the sum of cubes of primes and an almost prime,
Arch. Math. 69 (1997), no. 1, 13--19.

Zun Shan and Jia Hai Kan, On the representation of a large even integer as the
sum of a prime and an almost prime: the prime belongs to a fixed arithmetic
progression (Chinese. English, Chinese summary),
Acta Math. Sinica 40 (1997), no. 4, 625--638. Review by Tian Ze Wang  Link .

 Published in 1996 
Jiahai Kan and Zun Shan, On the divisor function d(n),
Mathematika 43 (1996), no. 2, 320--322 (1997).

 Published in 1994 
Jie Wu, Sur l'équation p + 2 = P2 dans les petits intervalles (French), [On the 
equation p + 2 = P2 in short intervals], J. London Math. Soc. (2) 49 (1994), 
no. 1, 61--72.  A similar result has been published by Saverio Salerno and 
 Antonio Vitolo  in 1993. This result has been improved by Ying Chun Cai 
 and Ming Gao Lu in 1999.  Review by A. Perelli  Link .

 Published in 1993 
Saverio Salerno and Antonio Vitolo,  p + 2 = P2  in short intervals, Note Mat. 
 13 (1993), no. 2, 309--328.  A similar result has been published by Jie Wu  
 in 1994. This result has been improved by Ying Chun Cai and Ming Gao Lu 
 in 1999.  Review by John B. Friedlander  Link .

 Published in 1990 
M. D. Coleman, On the equation b1 p - b2 P2 = b3,
J. Reine Angew. Math. 403 (1990), 1--66.

Jia Hai Kan, On the number of solutions of  p+h=Pr  ,
Math. Z. 203 (1990), no. 1, 37--42.

Jie Wu, Sur la suite des nombres premiers jumeaux. (French),
[On the series of twin primes], Acta Arith. 55 (1990), no. 4, 365--394.
  Important review by Wen-Bin Zhang   Link .

 Published in 1989 
É. Fouvry and F. Grupp, Weighted sieves and twin prime type equations,
Duke Math. J. 58 (1989), no. 3, 731--748.
 This paper establishes Chen's theorem with another proof. 

 Published in 1986 
Xiong Shao, Lower bounds for the number of solutions of $N-p=P\sb 3$. 
(Chinese. English summary), J. Math. (Wuhan) 6 (1986), no. 3, 307--314.

 Published in 1985 
Glyn Harman, Diophantine approximation with almost-primes and sums of two 
squares, Mathematika 32 (1985), no. 2, 301--310. 

 Published in 1984 
Mireille Car, Le théorème de Chen pour Fq[X] (French) [Chen's theorem 
for Fq[X] ,Dissertationes Math. (Rozprawy Mat.) 223 (1984), 54 pp.

Eugene K.-S. Ng, On the sequences N - p, p + 2 and the parity problem,
Arch. Math. 42 (1984), no. 5, 430--438.

 Published in 1983 
Sheng Gang Xie, The generalized twin prime problem (Chinese),
Adv. in Math. (Beijing) 12 (1983), no. 4, 313--320.

Sheng Gang Xie, On the $k$-twin primes problem, Acta Math. Sinica 26 (1983), no. 3, 378--384. 

 Published in 1982 
Jürgen G. Hinz, On the representation of even integers as sums of two almost 
primes in algebraic number fields, Mathematika 29 (1982), no. 1, 93--108.

 Published in 1979 
Cheng Dong Pan and Xia Xi Ding, A new mean value theorem,
Sci. Sinica 1979, Special Issue II on Math., 149--161.

 Published in 1978 
Jing Run Chen, On the representation of a large even integer as the sum of a prime 
and the product of at most two primes II, Sci. Sinica 21 (1978), no. 4, 421--430.

P. M. Ross, A short intervals result in additive prime number theory,
J. London Math. Soc. (2) 17 (1978), no. 2, 219--227.
 This paper contains an improvement of the paper published by the same author 
  in 1975,  which in turn contains a simplification of Chen's theorem. 
 Review by D. R. Heath-Brown:  Link .

 Published in 1977 
Akio Fujii, Some remarks on Goldbach's problem,
Acta Arith. 32 (1977), no. 1, 27--35.

 Published in 1975 
Jing Run Chen, On the distribution of almost primes in an interval, 
Sci. Sinica 18 (1975), no. 5, 611--627.

Cheng Dong Pan, Xia Xi Ding, and Yuan Wang, On the representation 
of every large even integer as a sum of a prime and an almost prime, 
Sci. Sinica 18 (1975), no. 5, 599--610.

P. M. Ross, On Chen's theorem that each large even number has the form ( p1 +  p2 ) 
or ( p1 +  p2 p3 ), J. London Math. Soc. (2) 10 (1975), no. 4, 500--506.
 This paper contains important simplifications of the proof Chen's theorem. 

 Published in 1973 
Jing Run Chen, On the representation of a larger even integer as the sum of a 
prime and the product of at most two primes, Sci. Sinica 16 (1973), 157-176.
 This is the historical paper establishing Chen's theorem. 
 Jing Run Chen announced his theorem in a paper published in 1966. 
See review by W. Schwarz:  Link .
 A simplification of Jing Run Chen's proof was published by P. M. Ross in 1975. 

 Published in 1966 
Jing Run 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.
 In this paper, Jing Run Chen states his famous theorem saying that both 
Goldbach's  Conjecture and the twin prime conjecture are "almost true". 
 He published the proof in his famous paper in 1973. 

Books with information about almost twin primes

Goldbach conjecture, Edited by Yuan Wang. Series in Pure Mathematics 4,
World Scientific Publishing Co., Singapore, 1984, xi+311 pp. 

Cheng Dong Pan and Cheng Biao Pan, Gedebahe caixiang (Chinese) 
[Goldbach's conjecture], Chuncui Shuxue yu Yingyong Shuxue Zhuanzhu 
[Series of Monographs in Pure and Applied Mathematics] 7, 
Kexue Chubanshe (Science Press), Beijing, 1981. vii+330 pp. 

Jean-Marc Deshouillers, Progrès récents des petits cribles arithmétiques [d'après 
Jing Run Chen, Henryk Iwaniec,$\ldots $] (French), 
Séminaire Bourbaki, 30e année (1977/78), Exp. No. 520, pp. 248--262, 
Lecture Notes in Math. 710, Springer, Berlin, 1979.

Yoichi Motohashi, Goldbach conjecture (Japanese), Seminaro Nota 
[Seminar Notes] 2, Ryukyu University, Naha, 1979. iii+112 pp.

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