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Chen primes
A prime number  $p$  is a Chen prime if  $p+2$  is either a prime or a semiprime.

Jing Run Chen, after which they are named, proved in 1966 that there are infinite such primes.

Binbin Zhou has proved in 2009 that the Chen primes contain arbitrarily long arithmetic progressions.

The smallest 3 × 3 magic square whose entries are Chen primes is

4711317
295989
101571

The first Chen primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257 more terms

Chen primes can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 13 101 181 + 99986539 aban 11 13 17 19 + 99000953 alt.fact. 19 101 4421 alternating 23 29 41 47 + 89898929 amenable 13 17 29 37 + 99999721 apocalyptic 157 251 443 499 + 29927 arithmetic 11 13 17 19 + 9999991 balanced p. 53 157 211 257 + 99999617 Bell 877 bemirp 1061 1091 1601 1901 + 19880981 c.decagonal 11 31 101 211 + 99971561 c.heptagonal 71 197 953 1471 + 99785911 c.pentagonal 31 181 1381 11731 + 99335281 c.square 13 41 113 181 + 99531941 c.triangular 19 31 109 199 + 99988591 Carol 47 16127 congruent 13 23 29 31 + 9999991 constructible 17 257 65537 Cunningham 17 31 37 101 + 99720197 Curzon 29 41 53 89 + 99996821 cyclic 11 13 17 19 + 9999991 d-powerful 89 379 2179 2243 + 9979337 de Polignac 127 149 251 337 + 99999839 deficient 11 13 17 19 + 9999991 dig.balanced 11 19 37 41 + 67084289 economical 11 13 17 19 + 19999739 emirp 13 17 31 37 + 99999827 equidigital 11 13 17 19 + 19999739 esthetic 23 67 89 101 + 67678789 Eulerian 11 65519 478271 evil 17 23 29 53 + 99999971 fibodiv 19 47 149 199 + 67645819 Fibonacci 13 89 233 514229 Friedman 127 347 12107 15641 + 995341 Gilda 29 683 2207 good prime 11 17 29 37 + 99927257 happy 13 19 23 31 + 9999991 hex 19 37 127 631 + 97179517 Hogben 13 31 157 211 + 99790111 Honaker 131 263 1039 1049 + 99973061 hungry 17 2003 iban 11 17 23 41 + 777317 iccanobiF 13 4139 idoneal 13 37 inconsummate 431 443 461 491 + 999149 Jacobsthal 11 683 2731 43691 junction 101 107 109 113 + 99999257 katadrome 31 41 53 71 + 9875321 Kynea 23 66047 Leyland 17 32993 lonely 23 53 211 2179 + 1272749 Lucas 11 29 47 199 + 3010349 lucky 13 31 37 67 + 9999397 m-pointer 23 2111 11261 11621 + 14111131 magnanimous 11 23 29 41 + 48804809 metadrome 13 17 19 23 + 23456789 modest 13 19 23 29 + 99959999 Motzkin 127 15511 nialpdrome 11 31 41 53 + 99999971 nude 11 oban 11 13 17 19 + 983 odious 11 13 19 31 + 99999839 Ormiston 1913 1931 18397 19013 + 99999131 palindromic 11 101 131 181 + 9981899 palprime 11 101 131 181 + 9981899 pancake 11 29 37 67 + 99510779 panconsummate 11 23 31 37 + 1291 pandigital 11 19 partition 11 101 pernicious 11 13 17 19 + 9999991 Perrin 17 29 14197 43721 Pierpont 13 17 19 37 + 57395629 plaindrome 11 13 17 19 + 88888999 prime 11 13 17 19 + 99999971 primeval 13 37 107 113 + 10034579 Proth 13 17 41 113 + 99893249 repdigit 11 repfigit 19 47 197 1084051 repunit 13 31 127 157 + 99790111 self 31 53 211 233 + 99999827 self-describing 10153331 10322321 12103331 12193133 + 33322727 sliding 11 29 101 641 star 13 37 181 337 + 98050837 straight-line 4567 23456789 strobogrammatic 11 101 181 116911 + 69911669 strong prime 11 17 29 37 + 99999839 super-d 19 31 107 127 + 9999931 tetranacci 29 401 tribonacci 13 149 trimorphic 251 499 751 4999 281249 truncatable prime 13 17 23 29 + 99979337 twin 11 13 17 19 + 99999587 uban 11 13 17 19 + 98000059 Ulam 11 13 47 53 + 9999161 undulating 101 131 181 191 + 1616161 upside-down 19 37 1289 1559 + 99791311 weak prime 13 19 23 31 + 99999971 weakly prime 584141 971767 1062599 4393139 + 99778351 Wieferich 3511 Woodall 17 23 191 524287 590489 Zuckerman 11 zygodrome 11 11177 11777 22277 + 99955577