Prime quadruplet
Encyclopedia
A prime quadruplet is a set of four primes
of the form {p, p+2, p+6, p+8}. This represents the closest possible grouping of four primes larger than 3. The first 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. (This structure is necessary to ensure that none of the four primes is divisible by 2, 3 or 5). A prime quadruplet of this form is also called a prime decade.
Some sources also call {2, 3, 5, 7} or {3, 5, 7, 11} prime quadruplets, while some other sources exclude {5, 7, 11, 13}.
A prime quadruplet contains two pairs of twin prime
s and two overlapping prime triplet
s.
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, 26, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 .
the largest known prime quadruplet has 2401 digits. It was found by Norman Luhn on 20 August 2011 and starts with
p = 1367848532291 × 5591# / 35 − 1, where 5591# is a primorial
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 prime
s, 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 first few prime quintuplets with p+12 are:
{5, 7, 11, 13, 17}, {11, 13, 17, 19, 23}, {101, 103, 107, 109, 113}, {1481, 1483, 1487, 1489, 1493}, {16061, 16063, 16067, 16069, 16073}, {19421, 19423, 19427, 19429, 19433}, {21011, 21013, 21017, 21019, 21023}, {22271, 22273, 22277, 22279, 22283}, {43781, 43783, 43787, 43789, 43793}, {55331, 55333, 55337, 55339, 55343} ... .
The first prime quintuplets with p−4 are:
{7, 11, 13, 17, 19}, {97, 101, 103, 107, 109}, {1867, 1871, 1873, 1877, 1879}, {3457, 3461, 3463, 3467, 3469}, {5647, 5651, 5653, 5657, 5659}, {15727, 15731, 15733, 15737, 15739}, {16057, 16061, 16063, 16067, 16069}, {19417, 19421, 19423, 19427, 19429}, {43777, 43781, 43783, 43787, 43789}, {79687, 79691, 79693, 79697, 79699}, {88807, 88811, 88813, 88817, 88819} ... .
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.
If both p−4 and p+12 are prime then it becomes a prime sextuplet. The first few:
{7, 11, 13, 17, 19, 23}, {97, 101, 103, 107, 109, 113}, {16057, 16061, 16063, 16067, 16069, 16073}, {19417, 19421, 19423, 19427, 19429, 19433}, {43777, 43781, 43783, 43787, 43789, 43793}
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.
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.
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...
of the form {p, p+2, p+6, p+8}. This represents the closest possible grouping of four primes larger than 3. The first prime quadruplets are
{5, 7, 11
11 (number)
11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
, 13
13 (number)
13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....
}, {11, 13, 17
17 (number)
17 is the natural number following 16 and preceding 18. It is prime.In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70...
, 19
19 (number)
19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...
}, {101
101 (number)
101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...
, 103
103 (number)
103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...
, 107
107 (number)
107 is the natural number following 106 and preceding 108.-In mathematics:One hundred [and] seven is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime....
, 109
109 (number)
109 is the natural number following 108 and preceding 110.-In mathematics:One hundred [and] nine is the 29th prime number, and also a Chen prime. The previous prime is 107, making them both twin primes...
}, {191
191 (number)
191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...
, 193
193 (number)
193 is the natural number following 192 and preceding 194.-In mathematics:* 193 is an odd number* 193 is a centered 32-gonal number* 193 is a deficient number, as 1 is less than 193* 193 is a happy number* 193 is a lucky number...
, 197
197 (number)
197 is the natural number following 196 and preceding 198.-In mathematics:* 197 is an odd number* 197 is a prime number** 197 is a Chen prime** 197 is an Eisenstein prime with no imaginary part** 197 is a strong prime** 197 is a twin prime with 199...
, 199
199 (number)
199 is the natural number following 198 and preceding 200.-In mathematics:* 199 is an odd number* 199 is a centered triangular number* 199 is a centered 33-gonal number* 199 is a deficient number, as 1 is less than 199* 199 is a Lucas number...
}, {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. (This structure is necessary to ensure that none of the four primes is divisible by 2, 3 or 5). A prime quadruplet of this form is also called a prime decade.
Some sources also call {2, 3, 5, 7} or {3, 5, 7, 11} prime quadruplets, while some other sources exclude {5, 7, 11, 13}.
A prime quadruplet contains two pairs of 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 and two overlapping prime triplet
Prime triplet
In mathematics, a prime triplet is a set of three prime numbers of the form or...
s.
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, 26, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 .
the largest known prime quadruplet has 2401 digits. It was found by Norman Luhn on 20 August 2011 and starts with
p = 1367848532291 × 5591# / 35 − 1, where 5591# is a primorial
Primorial
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...
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 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, 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
Ishango bone
The Ishango bone is a bone tool, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a sharp piece of quartz affixed to one end, perhaps for engraving...
although this is disputed.
Prime quintuplets
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:
{5, 7, 11, 13, 17}, {11, 13, 17, 19, 23}, {101, 103, 107, 109, 113}, {1481, 1483, 1487, 1489, 1493}, {16061, 16063, 16067, 16069, 16073}, {19421, 19423, 19427, 19429, 19433}, {21011, 21013, 21017, 21019, 21023}, {22271, 22273, 22277, 22279, 22283}, {43781, 43783, 43787, 43789, 43793}, {55331, 55333, 55337, 55339, 55343} ... .
The first prime quintuplets with p−4 are:
{7, 11, 13, 17, 19}, {97, 101, 103, 107, 109}, {1867, 1871, 1873, 1877, 1879}, {3457, 3461, 3463, 3467, 3469}, {5647, 5651, 5653, 5657, 5659}, {15727, 15731, 15733, 15737, 15739}, {16057, 16061, 16063, 16067, 16069}, {19417, 19421, 19423, 19427, 19429}, {43777, 43781, 43783, 43787, 43789}, {79687, 79691, 79693, 79697, 79699}, {88807, 88811, 88813, 88817, 88819} ... .
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.
If both p−4 and p+12 are prime then it becomes a prime sextuplet. The first few:
{7, 11, 13, 17, 19, 23}, {97, 101, 103, 107, 109, 113}, {16057, 16061, 16063, 16067, 16069, 16073}, {19417, 19421, 19423, 19427, 19429, 19433}, {43777, 43781, 43783, 43787, 43789, 43793}
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.
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.