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 (that is, if $\Omega (p+2)\le 2$ ,where $Ω$ is the Big Omega function). In 1966 Chen Jingrun proved that there are infinitely many such primes.

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)

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 2006, the largest known twin prime is 100314512544015 · 2¹⁷¹⁹⁶⁰ ± 1; it was found in 2006 by the Hungarians Zoltán Járai, Gabor Farkas, Timea Csajbok, Janos Kasza and Antal Járai. It has 51780 digits.

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