Prime quadruplet

A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {p, p+2, p+6, p+8}.[1] 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}.[2]

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 2011 the largest known prime quadruplet has 2401 digits.[3] It was found by Norman Luhn on 20 August 2011 and starts with

p = 1367848532291 × 5591# / 35 − 1, where 5591# is a primorial

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} %2B \frac{1}{7} %2B \frac{1}{11} %2B \frac{1}{13}\right) %2B \left(\frac{1}{11} %2B \frac{1}{13} %2B \frac{1}{17} %2B \frac{1}{19}\right) %2B \left(\frac{1}{101} %2B \frac{1}{103} %2B \frac{1}{107} %2B \frac{1}{109}\right) %2B \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.

Prime quintuplets

If {p, p+2, p+6, p+8} is a prime quadruplet and p−4 or p+12 is also prime, then the five primes form a prime quintuplet which is the closest admissible constellation of five primes. The first few prime quintuplets with p+12 are:

{5, 7, 11, 13, 17}, {11, 13, 17, 19, 23}, {101, 103, 107, 109, 113}, {1481, 1483, 1487, 1489, 1493}, {16061, 16063, 16067, 16069, 16073}, {19421, 19423, 19427, 19429, 19433}, {21011, 21013, 21017, 21019, 21023}, {22271, 22273, 22277, 22279, 22283}, {43781, 43783, 43787, 43789, 43793}, {55331, 55333, 55337, 55339, 55343} ... (sequence A022006 in OEIS).

The first prime quintuplets with p−4 are:

{7, 11, 13, 17, 19}, {97, 101, 103, 107, 109}, {1867, 1871, 1873, 1877, 1879}, {3457, 3461, 3463, 3467, 3469}, {5647, 5651, 5653, 5657, 5659}, {15727, 15731, 15733, 15737, 15739}, {16057, 16061, 16063, 16067, 16069}, {19417, 19421, 19423, 19427, 19429}, {43777, 43781, 43783, 43787, 43789}, {79687, 79691, 79693, 79697, 79699}, {88807, 88811, 88813, 88817, 88819} ... (sequence A022007 in OEIS).

A prime quintuplet contains two close pairs of twin primes, a prime quadruplet, and three overlapping prime triplets.

It is not known if there are infinitely many prime quintuplets. Once again, proving the twin prime conjecture might not necessarily prove that there are also infinitely many prime quintuplets. Also, proving that there are infinitely many prime quadruplets might not necessarily prove that there are infinitely many prime quintuplets.

If both p−4 and p+12 are prime then it becomes a prime sextuplet. The first few:

{7, 11, 13, 17, 19, 23}, {97, 101, 103, 107, 109, 113}, {16057, 16061, 16063, 16067, 16069, 16073}, {19417, 19421, 19423, 19427, 19429, 19433}, {43777, 43781, 43783, 43787, 43789, 43793}

Some 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.

A prime sextuplet contains two close pairs of twin primes, a prime quadruplet, four overlapping prime triplets, and two overlapping prime quintuplets.

It is not known if there are infinitely many prime sextuplets. Once again, proving the twin prime conjecture might not necessarily prove that there are also infinitely many prime sextuplets. Also, proving that there are infinitely many prime quintuplets might not necessarily prove that there are infinitely many prime sextuplets.

References

  1. ^ Weisstein, Eric W., "Prime Quadruplet" from MathWorld. Retrieved on 2007-06-15.
  2. ^ [1]
  3. ^ Tony Forbes. Prime k-tuplets. Retrieved on 2011-07-12.