A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. The even number 2p + 2 therefore satisfies Chen's theorem.
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, … A102540.
All of the supersingular primes are Chen primes.
Rudolf Ondrejka discovered the following 3x3 magic square of nine Chen primes:[1]
| 17 | 89 | 71 |
| 113 | 59 | 5 |
| 47 | 29 | 101 |
The lower member of a pair of twin primes is a Chen prime, by definition. In August 2009 Twin Prime Search and Primegrid found the largest known Chen prime, 65516468355 · 2333333 - 1 with 100355 digits.
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 progression.
References
External links
- The Prime Pages
- Green, Ben; Tao, Terence (2006). "Restriction theory of the Selberg sieve, with applications". Journal de théorie des nombres de Bordeaux 18 (1): 147–182. arΧiv:math.NT/0405581. http://www.emis.de/journals/JTNB/2006-1/jtnb18-1_english.html.
- Weisstein, Eric W., "Chen Prime" from MathWorld.
- The Chen primes contain arbitrarily long arithmetic progressions, Binbin Zhou, Acta Arith. 138 (2009), 301-315[1]
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