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Chen prime
From Wikipedia, the free encyclopedia
A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. In 1966 Chen Jingrun proved 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 OEIS)
The first few non-Chen primes are
43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241 (sequence A102540 in OEIS)
Note that all of the supersingular primes are Chen primes.
Rudolf Ondrejka (1928-2001) discovered the following 3x3 magic square of nine Chen primes[1]:
| 17 | 89 | 71 |
| 113 | 59 | 5 |
| 47 | 29 | 101 |
In October 2005 Micha Fleuren and PrimeForm e-group found the largest known Chen prime, (1284991359 · 298305 + 1) · (96060285 · 2135170 + 1) − 2 with 70301 digits.
The lower member of a pair of twin primes is a Chen prime, by definition. As of January 16, 2007, the largest known twin prime is 2003663613 · 2195000 ± 1.
Terence Tao and Ben Green proved in 2005 that there are infinitely many three-term arithmetic progressions of Chen primes.
- The Prime Pages
- Ben Green, Terence Tao, Restriction theory of the Selberg sieve, with applications
- Eric W. Weisstein, Chen Prime at MathWorld.
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