Quadruplet of prime numbers

A quadruplet of prime numbers is a group of four prime numbers, consisted of two pairs of Prime numbers twins separated by only three made up numbers consecutive, precisely, a multiple of 2, a multiple of 15 and another multiple of 2. Starting from the more small number first p of the quadruplet, the other prime numbers are p + 2, p + 6 and p + 8. The first small quadruplets of prime numbers is:

(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 exist two particular cases of quadruplets of prime numbers, which are not centered around one 15: (2, 3, 5, 7), and (5, 7,11,13).

One is unaware of if there exists an infinite number of quadruplets of prime numbers. To show the Conjecture of the prime numbers twins will not necessarily show that there exists also an infinity of quadruplets of prime numbers.

By using Mathematica, one can seek the multiples of 15 which center the quadruplets prime numbers with the following orders,

Select, PrimeQ * 15 - 4 && PrimeQ * 15 - 2 && PrimeQ * 15 + 2 && PrimeQ * 15 + 4 &]

% * 15

one can substitute another entirety for 10.000 in the Range function if it is wished.

One of largest the quadruplet of prime numbers known is centered around 10⁶⁹⁹ + 547634621255.

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