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Concept
Prime quadruplet
Summary
In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form {p,\ p+2,\ p+6,\ p+8}. This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.
The first eight 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}
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 are divisible by 2, 3 or 5). A prime quadruplet of this form is also called a prime decade.
A prime quadruplet can be described as a consecutive pair of twin primes, two overlapping sets of prime triplets, or two intermixed pairs of sexy primes.
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, 27, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 .
the largest known prime quadruplet has 10132 digits. It starts with p = 667674063382677 × 233608 − 1, found by Peter Kaiser.
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:
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.
Official source
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