Chen's theorem

In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes).

It is a weakened form of Goldbach's conjecture, which states that every even number is the sum of two primes.

History

The theorem was first stated by Chinese mathematician Chen Jingrun in 1966,^[1] with further details of the proof in 1973.^[2] His original proof was much simplified by P. M. Ross in 1975.^[3] Chen's theorem is a significant step towards Goldbach's conjecture, and a celebrated application of sieve methods.

Chen's theorem represents the strengthening of a previous result due to Alfréd Rényi, who in 1947 had shown there exists a finite K such that any even number can be written as the sum of a prime number and the product of at most K primes.^[4]^[5]

Variations

Chen's 1973 paper stated two results with nearly identical proofs.^[2]^: 158 His Theorem I, on the Goldbach conjecture, was stated above. His Theorem II is a result on the twin prime conjecture. It states that if h is a positive even integer, there are infinitely many primes p such that p + h is either prime or the product of two primes.

Ying Chun Cai proved the following in 2002:^[6]

There exists a natural number $N$ such that every even integer $n$ larger than $N$ is a sum of a prime less than or equal to $n^{0.95}$ and a number with at most two prime factors.

Tomohiro Yamada claimed a proof of the following explicit version of Chen's theorem in 2015:^[7]

Every even number greater than $e^{e^{36}}\approx 1.7\cdot 10^{1872344071119343}$ can be represented as the sum of a prime and a product of at most two primes.

In 2025, Daniel R. Johnston, Matteo Bordignon, and Valeriia Starichkova improved upon Yamada's result:^[8]

Every even number greater than $e^{e^{32.7}}\approx 1.4\cdot 10^{69057979807814}$ can be represented as the sum of a prime and a square-free number with at most two prime factors.

In addition in 2024, Bordignon and Starichkova^[9] showed that the bound can be lowered to $e^{e^{14}}\approx 2.5\cdot 10^{522284}$ assuming the Generalized Riemann hypothesis (GRH) for Dirichlet L-functions.

In 2019, Huixi Li gave a version of Chen's theorem for odd numbers. In particular, Li proved that every sufficiently large odd integer $N$ can be represented as^[10]

N=p+2a,

where $a$ has at most 2 prime factors. Here, the factor of 2 is necessitated since every prime (except for 2) is odd, causing $N-p$ to be even. Li's result can be viewed as an approximation to Lemoine's conjecture.

References

Citations

^ Chen, J.R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao. 11 (9): 385–386.
^ ^a ^b Chen, J.R. (1973). "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: 157–176.
^ Ross, P.M. (1975). "On Chen's theorem that each large even number has the form (p₁+p₂) or (p₁+p₂p₃)". J. London Math. Soc. Series 2. 10, 4 (4): 500–506. doi:10.1112/jlms/s2-10.4.500.
^ University of St Andrews - Alfréd Rényi
^ Rényi, A. A. (1948). "On the representation of an even number as the sum of a prime and an almost prime". Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya (in Russian). 12: 57–78.
^ Cai, Y.C. (2002). "Chen's Theorem with Small Primes". Acta Mathematica Sinica. 18 (3): 597–604. doi:10.1007/s101140200168. S2CID 121177443.
^ Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem". arXiv:1511.03409 [math.NT].
^ Johnston, Daniel R.; Bordignon, Matteo; Starichkova, Valeriia (2025-01-28). "An Explicit Version of Chen's Theorem". arXiv:2207.09452 [math.NT].
^ Bordignon, Matteo; Starichkova, Valeriia (2024). "An explicit version of Chen's theorem assuming the Generalized Riemann Hypothesis". The Ramanujan Journal. 64: 1213–1242. arXiv:2211.08844. doi:10.1007/s11139-024-00866-x.
^ Li, H. (2019). "On the representation of a large integer as the sum of a prime and a square-free number with at most three prime divisors". Ramanujan J. 49: 141–158.

Books

Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. Vol. 164. Springer-Verlag. ISBN 0-387-94656-X. Chapter 10.
Wang, Yuan (1984). Goldbach conjecture. World Scientific. ISBN 9971-966-09-3.

External links

Jean-Claude Evard, Almost twin primes and Chen's theorem
Weisstein, Eric W. "Chen's Theorem". MathWorld.

[1] Chen, J.R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao. 11 (9): 385–386.

[Chen_1973-2] Chen, J.R. (1973). "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: 157–176.

[3] Ross, P.M. (1975). "On Chen's theorem that each large even number has the form (p₁+p₂) or (p₁+p₂p₃)". J. London Math. Soc. Series 2. 10, 4 (4): 500–506. doi:10.1112/jlms/s2-10.4.500.

[4] University of St Andrews - Alfréd Rényi

[Alfréd_Rényi_1948-5] Rényi, A. A. (1948). "On the representation of an even number as the sum of a prime and an almost prime". Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya (in Russian). 12: 57–78.

[6] Cai, Y.C. (2002). "Chen's Theorem with Small Primes". Acta Mathematica Sinica. 18 (3): 597–604. doi:10.1007/s101140200168. S2CID 121177443.

[7] Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem". arXiv:1511.03409 [math.NT].

[8] Johnston, Daniel R.; Bordignon, Matteo; Starichkova, Valeriia (2025-01-28). "An Explicit Version of Chen's Theorem". arXiv:2207.09452 [math.NT].

[9] Bordignon, Matteo; Starichkova, Valeriia (2024). "An explicit version of Chen's theorem assuming the Generalized Riemann Hypothesis". The Ramanujan Journal. 64: 1213–1242. arXiv:2211.08844. doi:10.1007/s11139-024-00866-x.

[10] Li, H. (2019). "On the representation of a large integer as the sum of a prime and a square-free number with at most three prime divisors". Ramanujan J. 49: 141–158.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]