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}.[1] This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.
All prime quadruplets except {5, 7, 11, 13} are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19} for some integer n. (This structure is necessary to ensure that none of the four primes are divisible by 2,3 or 5). A prime quadruplet of this form is also called a prime decade.
All such prime decades have centers of form 210n + 15, 210n + 105, and 210n + 195 since the centers must be -1, O, 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 quints 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 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 11 or above also form the core of prime quintuplets and prime sextuplets by adding or subtracting 8 from their respective centers.
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. The number of prime quadruplets with n digits in base 10 for n = 2, 3, 4, ... is
1, 3, 7, 27, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 (sequence A120120 in the OEIS).
As of February2019[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:
B4 = 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 (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.
Excluding the first prime quadruplet, the shortest possible distance between two quadruplets {p, p + 2, p + 6, p + 8} and {q, q + 2, q + 6, q + 8} is q - p=30. The first occurrences of this are for p = 1006301, 2594951, 3919211, 9600551, 10531061, ... (OEIS:A059925).
If {p, p + 2, p + 6, p + 8} is a prime quadruplet and p− 4 or p + 12 is also prime, then the five primes form a prime quintuplet which is the closest admissible constellation of five primes. The first few prime quintuplets with p + 12 are:
A prime quintuplet contains two close pairs of twin primes, a prime quadruplet, and three overlapping prime triplets.
It is not known if there are infinitely many prime quintuplets. Once again, proving the twin prime conjecture might not necessarily prove that there are also infinitely many prime quintuplets. 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)).
Prime sextuplets
If both p− 4 and p + 12 are prime then it becomes a prime sextuplet. The first few:
Some sources also call {5, 7, 11, 13, 17, 19} a prime sextuplet. Our definition, all cases of primes {p− 4, p, p + 2, p + 6, p + 8, p + 12}, follows from defining a prime sextuplet as the closest admissible constellation of six primes.
A prime sextuplet contains two close pairs of twin primes, a prime quadruplet, four overlapping prime triplets, and two overlapping prime quintuplets.
All prime sextuplets except {7, 11, 13, 17, 19, 23} are of the form
for some integer n. (This structure is necessary to ensure that none of the six primes is divisible by 2, 3, 5 or 7).
It is not known if there are infinitely many prime sextuplets. Once again, proving the twin prime conjecture might not necessarily prove that there are also infinitely many prime sextuplets. 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)).
Prime quadruplets, quintuplets, and sextuplets are examples of prime constellations, and prime constellations are in turn examples of prime k-tuples. A prime constellation is a grouping of k primes, with minimum prime p and maximum prime p + n, meeting the following two conditions:
Not all residues modulo q are represented for any prime q
For any given k, the value of n is the minimum possible
More generally, a prime k-tuple occurs if the first condition but not necessarily the second condition is met.
Related Research Articles
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures.
In number theory, 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.
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.
90 (ninety) is the natural number following 89 and preceding 91.
105 is the natural number following 104 and preceding 106.
109 is the natural number following 108 and preceding 110.
700 is the natural number following 699 and preceding 701.
600 is the natural number following 599 and preceding 601.
800 is the natural number following 799 and preceding 801.
900 is the natural number following 899 and preceding 901. It is the square of 30 and the sum of Euler's totient function for the first 54 positive integers. In base 10 it is a Harshad number. It is also the first number to be the square of a sphenic number.
10,000 is the natural number following 9,999 and preceding 10,001.
In number theory, a Wagstaff prime is a prime number of the form
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered k-gonal number contains k more dots than the previous layer.
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: 7 = 71, 9 = 32 and 64 = 26 are prime powers, while 6 = 2 × 3, 12 = 22 × 3 and 36 = 62 = 22 × 32 are not.
In mathematics, a prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. The even number 2p + 2 therefore satisfies Chen's theorem.
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes converges to a finite value known as Brun's constant, usually denoted by B2. Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods.
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).
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