# 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 . 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, $p<100$, 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 progression  . 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 and $p\#$ is a primorial.

Rudolf Ondrejka constructed this magic square using only Chen primes:

 $\begin{bmatrix}17&89&71\\ 113&59&5\\ 47&29&101\\ \end{bmatrix}$

The magic constant is 177.

## References

Title Chen prime ChenPrime 2013-03-22 16:04:19 2013-03-22 16:04:19 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Definition msc 11N05