# Chen prime

Named after Chen Jingrun 1973[1] Chen, J. R. 2, 3, 5, 7, 11, 13 A109611Chen primes: primes p such that p + 2 is either a prime or a semiprime

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. 2996863034895 × 21290000 − 1, with 388342 decimal digits, is the largest known Chen prime as of March 2018. Sum of the reciprocals of Chen primes converges.

## 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.

Binbin Zhou[3] proved that the Chen primes contain arbitrarily long arithmetic progressions, improving on an earlier proof of Green and Tao[4] establishing the result for arithmetic progressions of length 3.

## 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
3. ^ Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica 138:4 (2009), pp. 301–315.
4. ^ Ben Green and Terrence Tao, Restriction theory of the Selberg sieve, with applications, Journal de Théorie des Nombres de Bordeaux 18 (2006), pp. 147–182.