A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {p, p+2, p+6, p+8}. This represents the closest possible grouping of four primes larger than 3. 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} (sequence A007530 in OEIS)

All prime quadruplets except {5, 7, 11, 13} are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19} for some integer n. (This structure is necessary to ensure that none of the four primes is divisible by 2, 3 or 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}.

A prime quadruplet contains two pairs of twin primes and two overlapping prime triplets.

It is not known if there are infinitely many prime quadruplets. A proof that there are infinitely many would imply the twin prime conjecture, but it is consistent with current knowledge that there may be infinitely many pairs of twin primes and only finitely 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 April 2012 the largest known prime quadruplet has 3024 digits. It was found by Peter Kaiser and starts with

p = 43697976428649 × 29999 − 1.

The constant representing the sum of the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, denoted by B4, is the sum of the reciprocals of all prime quadruplets:

$B_4 = left(frac{1}{5} + frac{1}{7} + frac{1}{11} + frac{1}{13}right) + left(frac{1}{11} + frac{1}{13} + frac{1}{17} + frac{1}{19}right) + left(frac{1}{101} + frac{1}{103} + frac{1}{107} + frac{1}{109}right) + cdots$

with value:

B4 = 0.87058 83800 ± 0.00000 00005.

This constant should not be confused with the Brun's constant for cousin primes, prime pairs of the form (p, p + 4), which is also written as B4.

The prime quadruplet {11, 13, 17, 19} is alleged to appear on the Ishango bone although this is disputed.

### Other articles related to "prime quadruplet, prime, primes":

... If both p−4 and p+12 are prime then it becomes a prime sextuplet ... sources also call {5, 7, 11, 13, 17, 19} a prime sextuplet ... Our definition, all cases of primes {p-4, p, p+2, p+6, p+8, p+12}, follows from defining a prime sextuplet as the closest admissible constellation of six primes ...

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