Английская Википедия:Chen prime
Шаблон:Short description Шаблон:Infobox integer sequence 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
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, ... Шаблон:OEIS.
The first few non-Chen primes are
- 43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, … Шаблон:OEIS.
All of the supersingular primes are Chen primes.
Rudolf Ondrejka discovered the following 3 × 3 magic square of nine Chen primes:[1]
17 | 89 | 71 |
113 | 59 | 5 |
47 | 29 | 101 |
Шаблон:As of, the largest known Chen prime is 2996863034895 × 21290000 − 1, with 388342 decimal digits.
The sum of the reciprocals of Chen primes converges.Шаблон:Cn
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.
Green and Tao showed that the Chen primes contain infinitely many arithmetic progressions of length 3.[2] Binbin Zhou generalized this result by showing that the Chen primes contain arbitrarily long arithmetic progressions.[3]
Notes
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 PrimesШаблон:Dead link, Scienca Sinica 16, 157-176, 1973.
References
External links
- ↑ Шаблон:Cite web
- ↑ 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.
- ↑ Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica 138:4 (2009), pp. 301–315.