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Prime QuadrupletA 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 are two special cases of prime quadruplets, which are not centered around a multiple of 15: (2, 3, 5, 7), and (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. Using Mathematica, one can search for multiples of 15 that center prime quadruplets with the following commands, SelectRange[10000, PrimeQ* 15 - 4 && PrimeQ* 15 - 2 && PrimeQ* 15 + 2 && PrimeQ* 15 + 4 &] % * 15 substituting the 10000 in the Range function with another integer if one desires. One of the largest known prime quadruplets is centered around 10699 + 547634621255.
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