Chen prime

Named after Jing Run Chen 1973[1] Chen, J. R. 2, 3, 5, 7, 11, 13 A109611

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:[2]

 17 89 71 113 59 5 47 29 101

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 ${\displaystyle (1284991359\times 2^{98305}+1)\times (96060285\times 2^{135170}+1)-2}$
having 70301 decimal digits.[citation needed]

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.

Notes

1.^ Chen primes were first described by Yuan, W. On the Representation of Large Even Integers as a Sum of a Product of at Most 3 Primes and a Product of at Most 4 Primes, Scienca Sinica 16, 157-176, 1973.

References

1. ^ Chen, J. R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao. 17: 385–386.
2. ^ Prime Curios! page on 59