Author of publication OEIS index A109611 | Publication year 1973 First terms 2, 3, 5, 7, 11, 13 | |
A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2p + 2 therefore satisfies Chen's theorem.
The Chen primes are named after Chen Jingrun, who proved in 1966 that there are infinitely many such primes. This result would also follow from the truth of the twin prime conjecture.
The first few Chen primes are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, … (sequence A109611 in the OEIS).The first few Chen primes that are not the lower member of a pair of twin primes are
2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, ... (sequence A063637 in the OEIS).The first few non-Chen primes are
43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, … (sequence A102540 in the OEIS).All of the supersingular primes are Chen primes.
Rudolf Ondrejka discovered the following 3x3 magic square of nine Chen primes:
The lower member of a pair of twin primes is by definition a Chen prime. Thus, 3756801695685×2666669 − 1 (having 200700 decimal digits), found by Primegrid, represents the largest known Chen prime as of December 25, 2011.
The largest known Chen prime at that time which is not a twin prime was
having 70301 decimal digits.
Further results
Chen also proved the following generalization: For any even integer h, there exist infinitely many primes p such that p + h is either a prime or a semiprime.
Terence Tao and Ben Green proved in 2005 that there are infinitely many three-term arithmetic progressions of Chen primes. Recently, Binbin Zhou proved that the Chen primes contain arbitrarily long arithmetic progressions.