Prime triplet

Innumber theory,aprime tripletis a set of threeprime numbersin 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 $(p, p + 4, p + 6)$ .^[1]With the exceptions of(2, 3, 5)and(3, 5, 7),this is the closest possiblegroupingof three prime numbers, since one of every three sequentialodd numbersis a multiple of three, and hence not prime (except for 3 itself).

Examples

The first prime triplets (sequenceA098420in theOEIS) 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)

Subpairs of primes

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: $(p, p + 6)$ .

Higher-order versions

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 aprime quintuplet.

Aprime quadruplet $(p, p + 2, p + 6, p + 8)$ contains two overlapping prime triplets, $(p, p + 2, p + 6)$ and $(p + 2, p + 6,p + 8)$ .

Conjecture on prime triplets

Similarly to thetwin prime conjecture,it is conjectured that there are infinitely many prime triplets. The first knowngigantic primetriplet was found in 2008 by Norman Luhn and François Morain. The primes are $(p, p + 2, p + 6)$ with $p$ = 2072644824759 × 2³³³³³− 1.As of October 2020^[update]the largest knownprovenprime triplet contains primes with 20008 digits, namely the primes $(p, p + 2, p + 6)$ with $p$ = 4111286921397 × 2⁶⁶⁴²⁰− 1.^[2]

TheSkewes numberfor the triplet $(p, p + 2, p + 6)$ is 87613571, and for the triplet $(p, p + 4, p + 6)$ it is 337867.^[3]

References

^Chris Caldwell.The Prime Glossary: prime triplefrom thePrime Pages.Retrieved on 2010-03-22.
^The Top Twenty: Tripletfrom the Prime Pages. Retrieved on 2013-05-06.
^Tóth, László (2019)."On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood"(PDF).Computational Methods in Science and Technology.25(3): 143–148.doi:10.12921/cmst.2019.0000033.Retrieved10 November2019.

External links

[1] Chris Caldwell.The Prime Glossary: prime triplefrom thePrime Pages.Retrieved on 2010-03-22.

[2] The Top Twenty: Tripletfrom the Prime Pages. Retrieved on 2013-05-06.

[3] Tóth, László (2019)."On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood"(PDF).Computational Methods in Science and Technology.25(3): 143–148.doi:10.12921/cmst.2019.0000033.Retrieved10 November2019.

[1]

[2]

[3]

v t e Prime numberclasses
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n!\pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin
By property	Wieferich(pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1,\dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p,2 p + 1$ ) Cunningham ( $p,2 p \pm 1, 4 p \pm 3, 8 p \pm 7,...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3,...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers