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Chen's Theorem

Chen's theorem states that every sufficiently large even number N can be written as N=p+m, where p is a prime and m in P union P_2, with P denoting the set of primes and P_2 the set of semiprimes (Chen 1966, 1973). This is a proved approximation to the Goldbach conjecture, which requires m itself to be prime.

Chen also proved an analogous result for twin primes. For every positive even integer h, there are infinitely many primes p for which p+h is either a prime or a semiprime (Chen 1973). Taking h=2 shows that there are infinitely many Chen primes. It does not prove the twin prime conjecture, since p+2 is allowed to be a semiprime rather than a prime.

Chen primes also occur in arbitrarily long prime arithmetic progressions (Zhou 2009).

See also

Almost Prime, Chen Prime, Goldbach Conjecture, Prime Arithmetic Progression, Prime Number, Schnirelmann's Theorem, Semiprime, Twin Prime Conjecture, Twin Primes

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References

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 17, 385-386, 1966.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. I." Sci. Sinica 16, 157-176, 1973.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. Sinica 16, 421-430, 1978.Hardy, G. H. and Wright, E. M. "Unsolved Problems Concerning Primes." Appendix §3 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 415-416, 1979.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 297, 1996.Rivera, C. "Problems & Puzzles: Conjecture 002.-Chen's Conjecture." https://www.primepuzzles.net/conjectures/conj_002.htm.Ross, P. M. "On Chen's Theorem That Each Large Even Number Has the Form p_1+p_2 or p_1+p_2p_3." J. London Math. Soc. 10, 500-506, 1975.Veritasium. "We're 99.9% Sure This Pattern Is True, but No One Can Prove It." Jun. 14, 2026. https://www.youtube.com/watch?v=8HBDE-msUjw.Zhou, B. "The Chen Primes Contain Arbitrarily Long Arithmetic Progressions." Acta Arith. 138, 301-315, 2009. https://doi.org/10.4064/aa138-4-1.

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Chen's Theorem

Cite this as:

Weisstein, Eric W. "Chen's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChensTheorem.html

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