Chen prime

If for a prime numberMathworldPlanetmath p it holds that p+2 is either a prime or a semiprime, then p is called a Chen primeMathworldPlanetmath. The name was assigned by Ben Green and Terrence Tao in recognition of Chen’s theoremMathworldPlanetmath that every sufficiently large even numberMathworldPlanetmath can be written as the sum of a prime and a semiprime. To give two examples of Chen primes: 41 is a Chen prime since 43 is also a prime, but 43 is itself not a Chen prime because 45 has one factor too many to be a semiprime; 47 is a Chen prime since 49, the square of a prime, is a semiprime.

Chen Jingrun proved that there are infinitely many Chen primes, which could turn out to be a step towards proving the twin prime conjectureMathworldPlanetmath. Just looking at say, p<100, it would appear that there are more Chen primes than non-Chen primes. (The former are listed in A109611 of Sloane’s OEIS, the latter in A102540). However, counting up to 17107, there are 986 Chen primes and 986 non-Chen; after that, the density of Chen primes gradually thins.

In 2005, Green and Tao proved that there are infinitely many Chen primes in arithmetic progressionPlanetmathPlanetmath. Jens Kruse Andersen and friends found this example, in which each prime has more than 3000 base 10 digits each: ((3850324118+892819689n)2411#+1)(4787#+1)-2 where -1<n<3 and p# is a primorial.

Rudolf Ondrejka constructed this magic square using only Chen primes:


The magic constant is 177.


  • 1 J. Chen, “On the Representation of a Large Even Integer as the Sum of a Prime and the ProductPlanetmathPlanetmath of at Most Two Primes” Sci. Sinica 16, pp. 157 - 176 (1973)
  • 2 B. Green and T. Tao, “RestrictionPlanetmathPlanetmathPlanetmath theory of the Selberg sieve, with applications”, pp. 5, 14, 18 - 19, 21 (2005)
Title Chen prime
Canonical name ChenPrime
Date of creation 2013-03-22 16:04:19
Last modified on 2013-03-22 16:04:19
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 11N05