Chen prime
If for a prime number^{} $p$ it holds that $p+2$ is either a prime or a semiprime, then $p$ is called a Chen prime^{}. The name was assigned by Ben Green and Terrence Tao in recognition of Chen’s theorem^{} that every sufficiently large even number^{} 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 conjecture^{}. Just looking at say, $$, it would appear that there are more Chen primes than non-Chen primes. (The former are listed in A109611 of Sloane’s OEIS, the latter in A102540). However, counting up to 17107, there are 986 Chen primes and 986 non-Chen; after that, the density of Chen primes gradually thins.
In 2005, Green and Tao proved that there are infinitely many Chen primes in arithmetic progression^{}. Jens Kruse Andersen and friends found this example, in which each prime has more than 3000 base 10 digits each: $((3850324118+892819689n)2411\mathrm{\#}+1)(4787\mathrm{\#}+1)-2$ where $$ and $p\mathrm{\#}$ is a primorial.
Rudolf Ondrejka constructed this magic square using only Chen primes:
$$\left[\begin{array}{ccc}\hfill 17\hfill & \hfill 89\hfill & \hfill 71\hfill \\ \hfill 113\hfill & \hfill 59\hfill & \hfill 5\hfill \\ \hfill 47\hfill & \hfill 29\hfill & \hfill 101\hfill \end{array}\right]$$ |
The magic constant is 177.
References
- 1 J. Chen, “On the Representation of a Large Even Integer as the Sum of a Prime and the Product^{} of at Most Two Primes” Sci. Sinica 16, pp. 157 - 176 (1973)
- 2 B. Green and T. Tao, “Restriction^{} 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 |