| Named after | Chen Jingrun |
|---|---|
| Publication year | 1973[1] |
| Author of publication | Chen, J. R. |
| First terms | 2,3,5,7,11,13 |
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Inmathematics, aprime numberp is called aChen prime ifp + 2 is either a prime or aproduct of two primes (also called a semiprime). Theeven number 2p + 2 therefore satisfiesChen's theorem.
The Chen primes are named afterChen Jingrun, whoproved in 1966 that there areinfinitely many such primes. This result would also follow from the truth of thetwin prime conjecture as the lower member of a pair oftwin primes is by definition a Chen prime.
The first few Chen primes are
The first few Chen primes that are not the lower member of a pair of twin primes are
The first few non-Chen primes are
All of thesupersingular primes are Chen primes.
Rudolf Ondrejka discovered the following 3 × 3magic square of nine Chen primes:[2]
| 17 | 89 | 71 |
| 113 | 59 | 5 |
| 47 | 29 | 101 |
As of March 2018[update], the largest known Chen prime is2996863034895 × 21290000 − 1, with388342 decimal digits.
The sum of thereciprocals of Chen primesconverges.[citation needed]
Chen also proved the following generalization: For any evenintegerh, there exist infinitely many primesp such thatp + h is either a prime or asemiprime.
Ben Green andTerence Tao showed that the Chen primes contain infinitely manyarithmetic progressions of length 3.[3] Binbin Zhou generalized this result by showing that the Chen primes contain arbitrarily long arithmetic progressions.[4]