Prime k-tuple
Encyclopedia
In number theory
, a prime k-tuple is an ordered set of values (i.e. a vector
) representing a repeatable pattern of prime number
s. A k-tuple is represented as (a, b, ...) to represent any set of values (n + a, n + b, ...) for all values of n. In practice, 0 is usually used for the lowest value of the k-tuple. A prime k-tuple is one which may be used to represent patterns of prime numbers.
Several of the shortest k-tuples are known by other common names:
A prime k-tuple is sometimes referred to as an admissible k-tuple. In order for a k-tuple to be admissible, it must not include the complete modulo
set of residue classes (i.e. the values 0 through p − 1) of any prime p less than or equal to k. For example, the complete modulo residue of p = 3 is 0, 1, and 2, so the numbers in a k-tuple modulo 3 would have to include at most two of these values to be admissible; otherwise the resulting numbers would always include a multiple of 3 and therefore could not all be prime unless one of the numbers is 3 itself. Although (0, 2, 4) is not admissible it does produce the single set of primes, (3, 5, 7). Some inadmissible k-tuples have more than one all-prime solution. The smallest of these is (0, 2, 8, 14, 26), which has two solutions: (3, 5, 11, 17, 29) and (5, 7, 13, 19, 31) where all congruences (mod 5) are included in both cases.
The first few prime constellations are:
A prime constellation is sometimes referred to as a prime k-tuplet, but some authors reserve that term for instances which are not part of longer k-tuplets.
The first Hardy–Littlewood conjecture predicts that the asymptotic frequency of any prime constellation can be calculated. While the conjecture is unproven it is considered likely to be true.
of k.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a prime k-tuple is an ordered set of values (i.e. a vector
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
) representing a repeatable pattern of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s. A k-tuple is represented as (a, b, ...) to represent any set of values (n + a, n + b, ...) for all values of n. In practice, 0 is usually used for the lowest value of the k-tuple. A prime k-tuple is one which may be used to represent patterns of prime numbers.
Several of the shortest k-tuples are known by other common names:
| (0, 2) | twin prime Twin prime A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and... s |
| (0, 4) | cousin prime Cousin prime In mathematics, cousin primes are prime numbers that differ by four; compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six.... s |
| (0, 6) | sexy prime Sexy prime In mathematics, a sexy prime is a prime number that differs from another prime number by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6... s |
| (0, 2, 6), (0, 4, 6) | prime triplet Prime triplet In mathematics, a prime triplet is a set of three prime numbers of the form or... s |
| (0, 6, 12) | sexy prime triplets |
| (0, 2, 6, 8) | prime quadruplets |
| (0, 6, 12, 18) | sexy prime quadruplets |
| (0, 2, 6, 8, 12), (0, 4, 6, 10, 12) | quintuplet primes |
| (0, 4, 6, 10, 12, 16) | sextuplet primes |
A prime k-tuple is sometimes referred to as an admissible k-tuple. In order for a k-tuple to be admissible, it must not include the complete modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
set of residue classes (i.e. the values 0 through p − 1) of any prime p less than or equal to k. For example, the complete modulo residue of p = 3 is 0, 1, and 2, so the numbers in a k-tuple modulo 3 would have to include at most two of these values to be admissible; otherwise the resulting numbers would always include a multiple of 3 and therefore could not all be prime unless one of the numbers is 3 itself. Although (0, 2, 4) is not admissible it does produce the single set of primes, (3, 5, 7). Some inadmissible k-tuples have more than one all-prime solution. The smallest of these is (0, 2, 8, 14, 26), which has two solutions: (3, 5, 11, 17, 29) and (5, 7, 13, 19, 31) where all congruences (mod 5) are included in both cases.
Prime constellations
An admissible prime k-tuple with the smallest possible maximum value s is a prime constellation. For all n ≥ k this will always produce consecutive primes.The first few prime constellations are:
| k | s | Constellation | smallest |
|---|---|---|---|
| 2 | 2 | (0, 2) | (3, 5) |
| 3 | 6 | (0, 2, 6) (0, 4, 6) | (5, 7, 11) (7, 11, 13) |
| 4 | 8 | (0, 2, 6, 8) | (5, 7, 11, 13) |
| 5 | 12 | (0, 2, 6, 8, 12) (0, 4, 6, 10, 12) | (5, 7, 11, 13, 17) (7, 11, 13, 17, 19) |
| 6 | 16 | (0, 4, 6, 10, 12, 16) | (7, 11, 13, 17, 19, 23) |
| 7 | 20 | (0, 2, 6, 8, 12, 18, 20) (0, 2, 8, 12, 14, 18, 20) | (11, 13, 17, 19, 23, 29, 31) (5639, 5641, 5647, 5651, 5653, 5657, 5659) |
| 8 | 26 | (0, 2, 6, 8, 12, 18, 20, 26) (0, 2, 6, 12, 14, 20, 24, 26) (0, 6, 8, 14, 18, 20, 24, 26) | (11, 13, 17, 19, 23, 29, 31, 37) (17, 19, 23, 29, 31, 37, 41, 43) (88793, 88799, 88801, 88807, 88811, 88813, 88817, 88819) |
| 9 | 30 | (0, 2, 6, 8, 12, 18, 20, 26, 30) (0, 4, 6, 10, 16, 18, 24, 28, 30) (0, 2, 6, 12, 14, 20, 24, 26, 30) (0, 4, 10, 12, 18, 22, 24, 28, 30) | (11, 13, 17, 19, 23, 29, 31, 37, 41) (13, 17, 19, 23, 29, 31, 37, 41, 43) (17, 19, 23, 29, 31, 37, 41, 43, 47) (88789, 88793, 88799, 88801, 88807, 88811, 88813, 88817, 88819) |
A prime constellation is sometimes referred to as a prime k-tuplet, but some authors reserve that term for instances which are not part of longer k-tuplets.
The first Hardy–Littlewood conjecture predicts that the asymptotic frequency of any prime constellation can be calculated. While the conjecture is unproven it is considered likely to be true.
Prime arithmetic progressions
A prime k-tuple of the form (0, n, 2n, ...) is said to be a prime arithmetic progression. In order for such a k-tuple to meet the admissibility test, n must be a multiple of the primorialPrimorial
In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied...
of k.