Prime quadruplet

From Free net encyclopedia

---

Current revision

A prime quadruplet is a group of four primes, consisting of two pairs of twin primes separated only by three non-primes, specifically, a multiple of 2, a multiple of 15 and another multiple of 2. From the smallest prime p of the quadruplet, the other primes are p + 2, p + 6 and p + 8. The first few prime quadruplets are

{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, 18429}, {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}, {97481, 97483, 97487, 97489}, {99131, 99133, 99137, 99139}

There is one special case of a prime quadruplet, which is not centered on a multiple of 15: {5, 7, 11, 13}. All other prime quadruplets are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19}.

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.

One of the largest known prime quadruplets is centered on 10⁶⁹⁹ + 547634621255.

The constant representing the sum the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, is approximately 0.87058.

The first and third members of a prime quadruplet are obviously Chen primes, but not quite so obviously, the second member of a prime quadruplet is never a Chen prime except for the very first quadruplet and the special quadruplet. The fourth member of a prime quadruplet is never a Stern prime.de:Primzahlvierling fr:Quadruplet de nombres premiers ja:四つ子素数 pl:Liczby czworacze