Math 2.0

Number theory Analytic number theory Theorem

Chen's theorem

Builds on Prime numberNumber of prime factors counted with multiplicityBombieri–Vinogradov theoremBrun sieve

Statement

Call a positive integer a P2P_2 number if it is a prime or a product of two primes. Equivalently, its number of prime factors counted with multiplicity is 11 or 22.

  1. There is an integer N0N_0 such that every even integer N>N0N > N_0 can be written as N=p+mN = p + m with pp prime and mm a P2P_2 number.
  2. There are infinitely many primes pp such that p+2p + 2 is a P2P_2 number.

Remarks

Both parts are due to Chen Jingrun. The proof is a weighted sieve argument. A lower bound sieve counts the m=N−pm = N - p with no small prime factor, and an upper bound sieve removes those with three or more prime factors. The Bombieri–Vinogradov theorem controls the remainder terms on average over the moduli. A switching argument, which counts the same configurations with the roles of the factors exchanged, is the key new step. Example: 100=3+97=7+93100 = 3 + 97 = 7 + 93, where 9797 is prime and 93=3⋅3193 = 3 \cdot 31 is a P2P_2 number. Part 1 is a weak form of Goldbach's conjecture, and part 2 a weak form of the twin prime conjecture.

Going deeper

For a student

Sieve methods cannot by themselves separate numbers with an odd number of prime factors from those with an even number; this is the parity problem. So a sieve can at best reach products of two primes, and Chen's theorem reaches exactly that limit. Earlier sieve results of the same kind, starting with the Brun sieve, allowed more prime factors on both sides.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptBig O notation, concept, no evidence levelBig O notationconceptBrun sieve, concept, no evidence levelBrun sieveconceptChebyshev functions, concept, no evidence levelChebyshev functionsconceptDedekind zeta function, concept, no evidence levelDedekind zeta functionconceptDirichlet L-function, concept, no evidence levelDirichlet L-functionconceptDirichlet character, concept, no evidence levelDirichlet characterconceptDivisibility, definition, no evidence levelDivisibilitydefinitionEuler's totient function, concept, no evidence levelEuler's totient funct…conceptLarge sieve, concept, no evidence levelLarge sieveconceptNumber of prime factors counted with multiplicity, concept, no evidence levelNumber of prime facto…conceptPrime number, concept, no evidence levelPrime numberconceptSiegel zero, concept, no evidence levelSiegel zeroconceptZero of a holomorphic function, concept, no evidence levelZero of a holomorphic…conceptvon Mangoldt function, concept, no evidence levelvon Mangoldt functionconceptGeneralized Riemann hypothesis, conjecture, no evidence levelGeneralized Riemann h…conjecturePrime number, definition, no evidence levelPrime numberdefinitionPrime-counting function, definition, no evidence levelPrime-counting functi…definitionPrime number theorem, statement, no evidence levelPrime number theoremstatementSiegel–Walfisz theorem, statement, no evidence levelSiegel–Walfisz theoremstatementBombieri–Vinogradov theorem, statement, no evidence levelBombieri–Vinogradov t…statementChen's theorem, statement, no evidence levelChen's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number field concept
  2. Big O notation concept
  3. Brun sieve concept
  4. Chebyshev functions concept
  5. Dedekind zeta function concept
  6. Dirichlet L-function concept
  7. Dirichlet character concept
  8. Divisibility definition
  9. Euler's totient function concept
  10. Large sieve concept
  11. Number of prime factors counted with multiplicity concept
  12. Prime number concept
  13. Siegel zero concept
  14. Zero of a holomorphic function concept
  15. von Mangoldt function concept
  16. Generalized Riemann hypothesis conjecture
  17. Prime number definition
  18. Prime-counting function definition
  19. Prime number theorem statement
  20. Siegel–Walfisz theorem statement
  21. Bombieri–Vinogradov theorem statement
  22. Chen's theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Chen's_theorem
About this page
TagWE0RQTR68H
URIm2://WE0RQTR68H
Kindstatement, theorem
WritingW0 stub
MSC11P32, 11N36
StatusThis copy of the site has no build data. Run m2 status WE0RQTR68H.
AboutGoldbach's conjecture, Twin prime conjecture
UsesPrime number, Number of prime factors counted with multiplicity, Bombieri–Vinogradov theorem, Brun sieve
Same as
Changed2 November 2026

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