Chen Prime

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

(https://primes.utm.edu/primes/page.php?id=75857), which has 70301 digits.

There are infinitely many cases of 3 Chen primes in arithmetic progression (Green and Tao 2005). The following 3074-digit case produces Chen primes for , 1, 2, where  denotes the primorial:

REFERENCES:

Andersen, J. K. "Chen AP3 with 3074 digits." https://groups.yahoo.com/group/primeform/message/6381.

Andersen, J. K. "Chen Prime with 70301 digits." https://groups.yahoo.com/group/primeform/message/6481.

Chen, J. R. "On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two Primes." Scientia Sinica 16, 157-176, 1973.

Evard, J.-C. "Almost Twin Primes and Chen's Theorem." https://www.math.utoledo.edu/~jevard/Page015.htm.

Green, B. and Tao, T. "Restriction Theory of the Selberg Sieve, with Applications." J. Théor. nombres Bordeaux 18, 147-182, 2006.

Sloane, N. J. A. Sequences A102540 and A109611 in "The On-Line Encyclopedia of Integer Sequences."