Chen's theorem

The statue of Chen Jingrun at Xiamen University.

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 giant step towards the Goldbach's conjecture, and a remarkable result of the 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 ${\displaystyle e^{e^{36}}\approx 1.7\cdot 10^{1872344071119343}}$ can be represented as the sum of a prime and a product of at most two primes.

In 2022, Matteo Bordignon found multiple errors in Yamada's proof, and provided an alternative proof for a lower bound:^[8]

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

Also in 2022, Bordignon and Valeriia Starichkova ^[9] showed that the bound can be lowered to ${\displaystyle e^{e^{15.85}}\approx 3.6\cdot 10^{3321634}}$ assuming the Generalized Riemann hypothesis (GRH) for Dirichlet L-functions. In 2024, Bordignon and Starichkova ^[10] improved this result by lowering the bound to ${\displaystyle e^{e^{14}}\approx 2.5\cdot 10^{522284}}$.

References

Citations

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.
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
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. Bordignon, Matteo (2022-02-08). "An Explicit Version of Chen's Theorem". Bulletin of the Australian Mathematical Society. 105 (2). Cambridge University Press (CUP): 344–346. arXiv:2207.09452. doi:10.1017/s0004972721001301. ISSN 0004-9727.
9. Bordignon, Matteo; Starichkova, Valeriia (2022). "An explicit version of Chen's theorem assuming the Generalized Riemann Hypothesis". arXiv:2211.08844.
10. Bordignon, Matteo; Starichkova, Valeriia (2024). "An explicit version of Chen's theorem assuming the Generalized Riemann Hypothesis". The Ramanujan Journal. 64: 1213–1242. doi:10.1007/s11139-024-00866-x.

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.