Prime quadruplet
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A prime quadruplet (sometimes called prime quadruple) is four primes of the form {p, p+2, p+6, p+8}. It is the closest four primes above 3 can be together. The first prime quadruplets are
{5, 7, 11, 13}, {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829}, {1481, 1483, 1487, 1489}, {1871, 1873, 1877, 1879}, {2081, 2083, 2087, 2089}, {3251, 3253, 3257, 3259}, {3461, 3463, 3467, 3469}, {5651, 5653, 5657, 5659}, {9431, 9433, 9437, 9439}, {13001, 13003, 13007, 13009}, {15641, 15643, 15647, 15649}, {15731, 15733, 15737, 15739}, {16061, 16063, 16067, 16069}, {18041, 18043, 18047, 18049}, {18911, 18913, 18917, 18919}, {19421, 19423, 19427, 19429}, {21011, 21013, 21017, 21019}, {22271, 22273, 22277, 22279}, {25301, 25303, 25307, 25309}, {31721, 31723, 31727, 31729}, {34841, 34843, 34847, 34849}, {43781, 43783, 43787, 43789}, {51341, 51343, 51347, 51349}, {55331, 55333, 55337, 55339}, {62981, 62983, 62987, 62989}, {67211, 67213, 67217, 67219}, {69491,69493, 69497, 69499}, {72221, 72223, 72227, 72229}, {77261, 77263, 77267, 77269}, {79691, 79693, 79697, 79699}, {81041, 81043, 81047, 81049}, {82721, 82723, 82727, 82729}, {88811, 88813, 88817, 88819}, {97841, 97843, 97847, 97849}, {99131,99133, 99137, 99139},
All prime quadruplets except {5, 7, 11, 13} are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19} (this is necessary to avoid the prime factors 2, 3 and 5). A prime quadruplet of this form is also called a prime decade.
Some sources also call {2, 3, 5, 7} or {3, 5, 7, 11} prime quadruplets, while some other sources exclude {5, 7, 11, 13}. Our definition, all cases of primes {p, p+2, p+6, p+8}, follows from defining a prime quadruplet as the closest admissible constellation of four primes. [1]
A prime quadruplet contains two close pairs of twin primes and two overlapping prime triplets.
It is not known if there are infinitely many prime quadruplets. Proving the twin prime conjecture might not necessarily prove that there are also infinitely many prime quadruplets. The number of prime quadruplets with n digits in base 10 for n = 2, 3, 4, ... is 1, 3, 7, 26, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 (sequence A120120 in OEIS).
As of 2006 the largest known prime quadruplet has 2058 digits. [2] It was found by Norman Luhn in 2005 and starts with
p = 4104082046 × 4799# + 5651, where 4799# is a primorial
The constant representing the sum of the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, is approximately 0.87058.
The prime quadruplet {11, 13, 17, 19} appears on the Ishango bone, one of the oldest artifacts from a civilization that used mathematics.
