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Chen prime

Prime number p where p+2 is prime or semiprime

Summary

Prime number p where p+2 is prime or semiprime

Field	Value
named_after	Chen Jingrun
publication_year	1973
author	Chen, J. R.
first_terms	2, 3, 5, 7, 11, 13
OEIS	A109611
OEIS_name	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, ... .

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, ... .

The first few non-Chen primes are

:43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, ... .

All of the supersingular primes are Chen primes.

Rudolf Ondrejka discovered the following 3 × 3 magic square of nine Chen primes:

47	29	101

, the largest known Chen prime is × 2 − 1, with decimal digits.

The sum of the reciprocals of Chen primes converges.

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. Binbin Zhou generalized this result by showing that the Chen primes contain arbitrarily long arithmetic progressions.

References

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.
"Prime Curios! 59".
[[Ben Green (mathematician). 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, [https://www.impan.pl/shop/publication/transaction/download/product/82619 The Chen primes contain arbitrarily long arithmetic progressions], ''[[Acta Arithmetica]]'' '''138''':4 (2009), pp. 301–315.

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

classes-of-prime-numbers 1966-in-science

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