Prime triplet

In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself). The first prime triplets are (5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647), (821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877, 881, 883), (881, 883, 887) A prime triplet contains a single pair of: Twin primes: (p, p + 2) or (p + 4, p + 6); Cousin primes: (p, p + 4) or (p + 2, p + 6); and Sexy primes: (p, p + 6). A prime can be a member of up to three prime triplets - for example, 103 is a member of (97, 101, 103), (101, 103, 107) and (103, 107, 109). When this happens, the five involved primes form a prime quintuplet. A prime quadruplet (p, p + 2, p + 6, p + 8) contains two overlapping prime triplets, (p, p + 2, p + 6) and (p + 2, p + 6, p + 8). Similarly to the twin prime conjecture, it is conjectured that there are infinitely many prime triplets. The first known gigantic prime triplet was found in 2008 by Norman Luhn and François Morain. The primes are (p, p + 2, p + 6) with p = 2072644824759 × 2^33333 − 1. the largest known proven prime triplet contains primes with 20008 digits, namely the primes (p, p + 2, p + 6) with p = 4111286921397 × 2^66420 − 1. The Skewes number for the triplet (p, p + 2, p + 6) is 87613571, and for the triplet (p, p + 4, p + 6) it is 337867.

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Prime k-tuple

In number theory, a prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a k-tuple (a, b, ...), the positions where the k-tuple matches a pattern in the prime numbers are given by the set of integers n such that all of the values (n + a, n + b, ...) are prime. Typically the first value in the k-tuple is 0 and the rest are distinct positive even numbers. Several of the shortest k-tuples are known by other common names: OEIS sequence covers 7-tuples (prime septuplets) and contains an overview of related sequences, e.

Sexy prime

In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and 11 − 5 = 6. The term "sexy prime" is a pun stemming from the Latin word for six: sex. If p + 2 or p + 4 (where p is the lower prime) is also prime, then the sexy prime is part of a prime triplet.

Prime quadruplet

In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form {p,\ p+2,\ p+6,\ p+8}. This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. The first eight prime quadruplets are: {5, 7, 11, 13}, {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} All prime quadruplets except {5, 7, 11, 13} are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19} for some integer n.

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