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Theorem. Every sufficiently large even integer  can be expressed as the sum of two primes  , or the sum of a prime and a semiprime  , where  ,  and  are all distinct primes. “Sufficiently large” could mean  . For example, 62 can be represented as  in seven different ways:
 ,
 ,
 , etc.
This theorem was proven by Chen Jingrun in 1966 but had to delay publishing his results until 1973 because of political problems in his native China. Chen's proof has been considered “a highly technical application of sieving methods.” (Eisenstein et al, 2004) Ross simplified Chen's proof almost a decade later. Still, a summary of the proof can run for dozens of pages. A much shorter, but excessively broad summary that can fit in here goes something like this: reduction to sieving, estimation of sieving functions, search for upper bounds using the Jurkat-Richert theorem, using a bilinear form inequality, and joining together of all these results to create a function that counts the number of representations of a given number as either  or  , and showing that that function always returns a positive integer when the given number is sufficiently large.
The Zumkeller-Lebl conjecture, an attempt to generalize Chen's theorem to odd numbers, and unproven as of 2008, states that sufficiently large integers, be they even or odd, can also be represented as  . To represent odd numbers this way, only one of the primes can be 2 (or both  ).
Levy's conjecture, which applies only to odd numbers, has  and  and  both odd primes.
Sequence A100952 of Sloane's OEIS lists the known twenty-one small integers that can't be expressed as specified by the theorem.
- 1
- J. R. 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.
- 2
- E. Eisenstein, L. Jain, A. Felix, ``A summary of the proof of Chen's theorem''. Ann Arbor: University of Michigan (2004)
- 3
- P. M. Ross, ``On Chen's theorem that each large even number has the form
 or
 ,'' J. London Math. Soc. 10 (1975), 500 - 506
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"Chen's theorem" is owned by PrimeFan. [ full author list (2) | owner history (1) ]
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Cross-references: OEIS, sequence, Levy's conjecture, represent, odd, even, odd numbers, conjecture, integer, positive, representations, number, inequality, bilinear form, upper bounds, functions, reduction, proof, Chen Jingrun, mean, semiprime, primes, sum, even integer
There are 3 references to this entry.
This is version 6 of Chen's theorem, born on 2006-10-13, modified 2008-08-30.
Object id is 8456, canonical name is ChensTheorem.
Accessed 1284 times total.
Classification:
| AMS MSC: | 11N05 (Number theory :: Multiplicative number theory :: Distribution of primes) |
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Pending Errata and Addenda
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