A Chen prime is a prime number for which is either a prime or semiprime.
Chen primes are named after Jing Run Chen who proved in 1966 that there are infinitely
many such primes (Chen's theorem).
The first Chen primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (OEIS A109611). The first primes that are not Chen primes are 43, 61, 73, 79, 97, 103, 151, ... (OEIS
A102540).
The lesser of any twin prime is always a Chen prime. Apart from twin prime records, the largest known Chen
prime known as of Oct. 2005 was
Green and Tao (2006) proved that there are infinitely many three-term arithmetic progressions of Chen primes. Zhou (2009) strengthened this result by proving that
the Chen primes contain arbitrarily long prime
arithmetic progressions. The following 3074-digit case produces Chen primes for
, 1, 2, where denotes the primorial:
for which 
is either a prime or semiprime.
Chen primes are named after Jing Run Chen who proved in 1966 that there are infinitely
many such primes (Chen's theorem).
, 1, 2, where 
denotes the primorial:
![[(3850324118+892819689×n)×2411#+1]×(4787#+1)-2.](/images/equations/ChenPrime/NumberedEquation2.svg)
