Prime quadruplet
In number theory, a prime quadruplet (sometimes called a prime quadruple) is a set of four prime numbers of the form Template:Math<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:PrimeQuadruplet%7CPrimeQuadruplet.html}} |title = Prime Quadruplet |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }} Retrieved on 2007-06-15.</ref> This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.
Prime quadrupletsEdit
The first eight prime quadruplets are:
Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap(sequence A007530 in the OEIS)
All prime quadruplets except Template:Nowrap are of the form Template:Math for some integer Template:Mvar. 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 end with the digit 1 and the last prime ends with the digit 9, in base 10. Thus 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 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. The number of prime quadruplets with Template:Mvar digits in base 10 for Template:Nowrap is
- 1, 3, 7, 27, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 (sequence A120120 in the OEIS).
Template:As of the largest known prime quadruplet has 10132 digits.<ref>The Top Twenty: Quadruplet at The Prime Pages. Retrieved on 2019-02-28.</ref> It starts with Template:Nowrap, found by Peter Kaiser.
The constant representing the sum of the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, denoted by Template:Math, is the sum of the reciprocals of all prime quadruplets:
<math display=block>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</math>
with value:
- Template:Math = 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 Template:Math, which is also written as Template:Math.
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 Template:Math and Template:Math is Template:Math = 30. The first occurrences of this are for Template:Mvar = 1006301, 2594951, 3919211, 9600551, 10531061, ... (Template:OEIS2C).
The Skewes number for prime quadruplets Template:Math is 1172531 (Template:Harvtxt).
Prime quintupletsEdit
If Template:Math is a prime quadruplet and Template:Math or Template:Math 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 Template:Math are:
- Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap … Template:OEIS2C.
The first prime quintuplets with Template:Math are:
- Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap Template:Nowrap... Template:OEIS2C.
A prime quintuplet contains two close pairs of twin primes, a prime quadruplet, and three overlapping prime triplets. The first prime of a quintuplet starting above 5 will end with the digit 1 or 7 in base 10 and the last prime will end with the digit 3 or 9.
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 Template:Math is 21432401 (Template:Harvtxt).
Prime sextupletsEdit
If both Template:Math and Template:Math are prime then it becomes a prime sextuplet. The first few:
Some sources also call Template:Nowrap a prime sextuplet. Our definition, all cases of primes Template:Math 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. The first prime of a sextuplet will end with the digit 7 in base 10 and the last prime will end with the digit 3.
All prime sextuplets except Template:Nowrap are of the form <math display=block>\{210n + 97,\ 210n + 101,\ 210n + 103,\ 210n + 107,\ 210n + 109,\ 210n + 113\}</math>for some integer Template:Mvar. (This structure is necessary to ensure that none of the six primes is divisible by Template:Nowrap).
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.
A prime sextuple is the largest k-tuple with spacing no greater than 4 between primes.
The Skewes number for the tuplet Template:Math is 251331775687 (Template:Harvtxt).
Prime k-tuplesEdit
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Prime quadruplets, quintuplets, and sextuplets are examples of prime constellations, and prime constellations are in turn examples of prime Template:Mvar-tuples. A prime constellation is a grouping of Template:Mvar primes, with minimum prime Template:Mvar and maximum prime Template:Math, meeting the following two conditions:
- Not all residues modulo Template:Mvar are represented for any prime Template:Mvar
- For any given Template:Mvar, the value of Template:Mvar is the minimum possible
More generally, a prime Template:Mvar-tuple occurs if the first condition but not necessarily the second condition is met.