Prime quadruplet

The first eight prime quadruplets are: {{nowrap|{5, 7, 11, 13},}} {{nowrap|{11, 13, 17, 19},}} {{nowrap|{101, 103, 107, 109},}} {{nowrap|{191, 193, 197, 199},}} {{nowrap|{821, 823, 827, 829},}} {{nowrap|{1481, 1483, 1487, 1489},}} {{nowrap|{1871, 1873, 1877, 1879},}} {{nowrap|{2081, 2083, 2087, 2089} }} All prime quadruplets except {{nowrap|{5, 7, 11, 13} }} are of the form {{math|{30n + 11, 30n + 13, 30n + 17, 30n + 19} }} for some integer . This structure is necessary to ensure that none of the four primes are divisible by 2, 3 or 5. The first prime of all such quadruplets ends with the digit 1 and the last prime ends with the digit 9, in base 10. Thus a prime quadruplet of this form is called a prime decade. A prime quadruplet can be described as a consecutive pair of twin primes, two overlapping sets of prime triplets, or two intermixed pairs of sexy primes. These "quad" primes can also form the core of prime quintuplets and prime sextuplets when adding or subtracting 8 from their centers yields a prime. All prime decades starting above 5 have centers of the form 210n + 15, 210n + 105, or 210n + 195, since the centers must be −1, 0, or 1, modulo 7. The +15 form may also give rise to a (high) prime quintuplet; the +195 form can also give rise to a (low) quintuplet; while the +105 form can yield both types of quintuplets and possibly prime sextuplets. It is no accident that each prime in a prime decade is displaced from its center by a power of 2, since all centers are odd and divisible by both 3 and 5. It is not known if there are infinitely many prime quadruplets. A proof that there are infinitely many would imply the twin prime conjecture, but it is consistent with current knowledge that there may be infinitely many pairs of twin primes and only finitely many prime quadruplets. However, it is conjectured that there are infinitely many prime quadruplets. The number of prime quadruplets with digits in base 10 for is :1, 3, 7, 27, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 . It is conjectured that the asymptotic formula for the number of prime quadruplets below is C \frac{x}{(\ln x)^4} , where C \sim 4.151180863 . the largest known prime quadruplet has 10132 digits. It starts with , found by Peter Kaiser. The constant representing the sum of the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, denoted by , 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}\right) + \left(\frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19}\right) + \left(\frac{1}{101} + \frac{1}{103} + \frac{1}{107} + \frac{1}{109}\right) + \cdots with value: : = 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 , which is also written as . The prime quadruplet {11, 13, 17, 19} is alleged to appear on the Ishango bone although this is disputed. Excluding the first prime quadruplet, the shortest possible distance between two quadruplets {{math|{p, p + 2, p + 6, p + 8} }} and {{math|{q, q + 2, q + 6, q + 8} }} is = 30. The first occurrences of this are for = 1006301, 2594951, 3919211, 9600551, 10531061, ... (). The Skewes number for prime quadruplets {{math|{p, p + 2, p + 6, p + 8} }} is 1172531 (). ==Prime quintuplets==