Chen prime

Chen prime
Named afterChen Jingrun
Publication year1973[1]
Author of publicationChen, J. R.
First terms2, 3, 5, 7, 11, 13
OEIS index
  • A109611
  • Chen primes: primes p such that p + 2 is either a prime or a semiprime

In mathematics, 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 as the lower member of a pair of twin primes is by definition a Chen prime.

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 3 × 3 magic square of nine Chen primes:[2]

17 89 71
113 59 5
47 29 101

As of March 2018, the largest known Chen prime is 2996863034895 × 21290000 − 1, with 388342 decimal digits.

The sum of the reciprocals of Chen primes converges.[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.

Ben Green and Terence Tao showed that the Chen primes contain infinitely many arithmetic progressions of length 3.[3] Binbin Zhou generalized this result by showing that the Chen primes contain arbitrarily long arithmetic progressions.[4]

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! 59". t5k.org. Retrieved 2023-12-13.
  3. ^ Ben Green and Terence Tao, Restriction theory of the Selberg sieve, with applications, Journal de Théorie des Nombres de Bordeaux 18 (2006), pp. 147–182.
  4. ^ Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica 138:4 (2009), pp. 301–315.