Prime quadruplet
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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), (97841, 97483, 97487, 97489), (99131, 99133, 99137, 99139)
There is one special case of a prime quadruplet, which is not centered around a multiple of 15: (5, 7, 11, 13).
It is not known if there are infinitely many prime quadruplets. Proving the twin prime conjecture might not necessarily prove that there also infinitely many prime quadruplets.
One of the largest known prime quadruplets is centered around 10699 + 547634621255.
The constant representing the sum the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, is approximately 0.87058.de:Primzahlquadruplet pl:Liczby czworacze