Chen prime

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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.

Note that 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

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 primes are 2003663613 · 2¹⁹⁵⁰⁰⁰ ± 1, with 58711 digits.

[edit] 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 product of two primes.

Terence Tao and Ben Green proved in 2005 that there are infinitely many three-term arithmetic progressions of Chen primes.

[edit] References

[edit] External links

• The Prime Pages
• Ben Green, Terence Tao, Restriction theory of the Selberg sieve, with applications
• Eric W. Weisstein, Chen Prime at MathWorld.