In number theory, ♥♥♥♥ primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both ♥♥♥♥ primes, because both are prime and 11 − 5 = 6.

The term "♥♥♥♥ prime" is a pun stemming from the Latin word for six: ♥♥♥.

If p + 2 or p + 4 (where p is the lower prime) is also prime, then the ♥♥♥♥ prime is part of a prime triplet. In August 2014 the Polymath group, seeking the proof of the twin prime conjecture, showed that if the generalized Elliott–Halberstam conjecture is proven, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6 and as such they are either twin, cousin or ♥♥♥♥ primes.[1]

Primorial n# notation
Further information: Primorial § Definition for natural numbers
As used in this article, n# stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ n.

Types of groupings
♥♥♥♥ prime pairs
The ♥♥♥♥ primes (sequences OEIS: A023201 and OEIS: A046117 in OEIS) below 500 are:

(5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107), (103,109), (107,113), (131,137), (151,157), (157,163), (167,173), (173,179), (191,197), (193,199), (223,229), (227,233), (233,239), (251,257), (257,263), (263,269), (271,277), (277,283), (307,313), (311,317), (331,337), (347,353), (353,359), (367,373), (373,379), (383,389), (433,439), (443,449), (457,463), (461,467).
As of April 2022, the largest-known pair of ♥♥♥♥ primes was found by S. Batalov and has 51,934 digits. The primes are:

p = 11922002779 x (2172486 - 286243) + 286245 - 5
p+6 = 11922002779 x (2172486 - 286243) + 286245 + 1[2]
♥♥♥♥ prime triplets
♥♥♥♥ primes can be extended to larger constellations. Triplets of primes (p, p+6, p+12) such that p+18 is composite are called ♥♥♥♥ prime triplets. Those below 1,000 are (OEIS: A046118, OEIS: A046119, OEIS: A046120):

(7,13,19), (17,23,29), (31,37,43), (47,53,59), (67,73,79), (97,103,109), (101,107,113), (151,157,163), (167,173,179), (227,233,239), (257,263,269), (271,277,283), (347,353,359), (367,373,379), (557,563,569), (587,593,599), (607,613,619), (647,653,659), (727,733,739), (941,947,953), (971,977,983).
In January 2005 Ken Davis set a record for the largest-known ♥♥♥♥ prime triplet with 5132 digits:

p = (84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1) / 35 + 1. [3]
In May 2019, Peter Kaiser improved this record to 6,031 digits:

p = 10409207693×220000−1.[4]
Gerd Lamprecht improved the record to 6,116 digits in August 2019:

p = 20730011943×14221#+344231.[5]
Ken Davis further improved the record with a 6,180 digit Brillhart-Lehmer-Selfridge provable triplet in October 2019:

p = (72865897*809857*4801#*(809857*4801#+1)+210)*(809857*4801#-1)/35+1[6]
Norman Luhn & Gerd Lamprecht improved the record to 6,701 digits in October 2019:

p = 22582235875×222224+1.[7]
Serge Batalov improved the record to 15,004 digits in April 2022:

p = 2494779036241x249800+1.[8]
♥♥♥♥ prime quadruplets
♥♥♥♥ prime quadruplets (p, p+6, p+12, p+18) can only begin with primes ending in a 1 in their decimal representation (except for the quadruplet with p = 5). The ♥♥♥♥ prime quadruplets below 1000 are (OEIS: A023271, OEIS: A046122, OEIS: A046123, OEIS: A046124):

(5,11,17,23), (11,17,23,29), (41,47,53,59), (61,67,73,79), (251,257,263,269), (601,607,613,619), (641,647,653,659).
In November 2005 the largest-known ♥♥♥♥ prime quadruplet, found by Jens Kruse Andersen had 1,002 digits:

p = 411784973 · 2347# + 3301.[9]
In September 2010 Ken Davis announced a 1,004 digit quadruplet with p = 23333 + 1582534968299.[10]

In May 2019 Marek Hubal announced a 1,138 digit quadruplet with p = 1567237911 × 2677# + 3301.[11][12]

In June 2019 Peter Kaiser announced a 1,534 digit quadruplet with p = 19299420002127 × 25050 + 17233.[13]

In October 2019 Gerd Lamprecht and Norman Luhn announced a 3,025 digit quadruplet with p = 121152729080 × 7019#/1729 + 1.[14]

♥♥♥♥ prime quintuplets
In an arithmetic progression of five terms with common difference 6, one of the terms must be divisible by 5, because 5 and 6 are relatively prime. However, all multiples of 5 (except itself) cannot be prime numbers. Thus, the only ♥♥♥♥ prime quintuplet is (5,11,17,23,29); no longer sequence of ♥♥♥♥ primes is possible, since adding 6 to the last number in the set of ♥♥♥♥ prime quintuplets (29) equals 35, which is a composite number.