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, actually 2 or 4, since all centers are odd and divisible by both 3 and 5.
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.
The number of prime quadruplets with n digits in base 10 for n = 2, 3, 4, ... is As of February 2019[update] the largest known prime quadruplet has 10132 digits.
[2] It starts with p = 667674063382677 × 233608 − 1, found by Peter Kaiser.
The constant representing the sum of the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, denoted by B4, is the sum of the reciprocals of all prime quadruplets:
with value: This constant should not be confused with the Brun's constant for cousin primes, prime pairs of the form (p, p + 4), which is also written as B4.
The prime quadruplet {11, 13, 17, 19} is alleged to appear on the Ishango bone although this is disputed.
The Skewes number for prime quadruplets {p, p + 2, p + 6, p + 8} is 1172531 (Tóth (2019)).
Also, proving that there are infinitely many prime quadruplets might not necessarily prove that there are infinitely many prime quintuplets.
The Skewes number for prime quintuplets {p, p + 2, p + 6, p + 8, p + 12} is 21432401 (Tóth (2019)).
The first few: Some sources also call {5, 7, 11, 13, 17, 19} a prime sextuplet.
for some integer n. (This structure is necessary to ensure that none of the six primes is divisible by 2, 3, 5 or 7).
Also, proving that there are infinitely many prime quintuplets might not necessarily prove that there are infinitely many prime sextuplets.
The Skewes number for the tuplet {p, p + 4, p + 6, p + 10, p + 12, p + 16} is 251331775687 (Tóth (2019)).