Prime quadruplet

A prime quadruplet (sometimes called prime quadruple) is four primes of the form {"p", "p"+2, "p"+6, "p"+8}. [MathWorld|urlname=PrimeQuadruplet|title=Prime Quadruplet Retrieved on 2007-06-15.] It is the closest four primes above 3 can be together, because one of the numbers {"p", "p"+2, "p"+4} is always divisible by 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}, {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 {30"n" + 11, 30"n" + 13, 30"n" + 17, 30"n" + 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}. The common definition given here, 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. [http://primes.utm.edu/glossary/page.php?sort=PrimeConstellation]

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 OEIS|id=A120120.

As of 2007 the largest known prime quadruplet has 2058 digits. [Tony Forbes. [http://anthony.d.forbes.googlepages.com/ktuplets.htm "Prime k-tuplets"] . Retrieved on 2007-09-01.] 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, denoted by "B"₄, is the sum of the reciprocals of all prime quadruplets:

:$B\_4\; =\; left(frac\{1\}\{5\}\; +\; frac\{1\}\{7\}\; +\; frac\{1\}\{11\}\; +\; frac\{1\}\{13\}\; ight)+\; left(frac\{1\}\{11\}\; +\; frac\{1\}\{13\}\; +\; frac\{1\}\{17\}\; +\; frac\{1\}\{19\}\; ight)+\; left(frac\{1\}\{101\}\; +\; frac\{1\}\{103\}\; +\; frac\{1\}\{107\}\; +\; frac\{1\}\{109\}\; ight)\; +\; cdots$

with value:

:"B"₄ = 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 "B"₄.

The prime quadruplet {11, 13, 17, 19} appears on the Ishango bone, one of the oldest artifacts from a civilization that used mathematics.

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 OEIS|id=A022006:

{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}

The first prime quintuplets with "p"−4 are (OEIS2C|id=A022007):

{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}

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