# prime k-tuplet

A prime *k*-tuple is a repeatable pattern of primes that are as
close as possible together (we will be more precise in a moment).
For example, twin primes are 2-tuples. Prime triplets are
3-tuples. These have the patterns {*p*, *p*+2, *p*+6}
or {*p*, *p*+4, *p*+6}.

4-tuples have the form {*p*, *p*+2, *p*+6, *p*+8}.
There is a pair of twin primes in every
prime 3-tuple, and a prime 3-tuple in every
prime 4-tuple (but not prime *k*-tuple
in every prime (*k*+1)-tuple, 7-tuples do not include 6-tuples). So
some authors use **prime k-tuplet** to mean a
prime

*k*-tuple which is not part of a prime (

*k*+1)-tuple. They would similarly distinguish

**prime triplet**from prime triple and

**prime quadruplet**from prime quadruple.

To make our definition precise we must first define the length
of the *k*-tuple. Let ** s(k)** to be the smallest number

*s*for which there exist

*k*integers

*b*

_{1}<

*b*

_{2}< ... <

*b*

_{k}, with

*b*

_{k}-

*b*

_{1}=

*s*and, for every prime

*q*, not all the residues modulo

*q*are represented by

*b*

_{1},

*b*

_{2}, ...,

*b*

_{k}. A

**prime**is then defined as a sequence of consecutive primes {

*k*-tuplet*p*

_{1},

*p*

_{2}, ...,

*p*

_{k}}such that for every prime

*q*, not all the residues modulo

*q*are represented by

*p*

_{1},

*p*

_{2}, ...,

*p*

_{k}, and

*p*

_{k}-

*p*

_{1}=

*s*(

*k*). This definition excludes a finite number (for each

*k*) of dense clusters at the beginning of the prime number sequence. For example, {97, 101, 103, 107, 109} satisfies the conditions of the definition of a prime 5-tuplet , but {3, 5, 7, 11, 13} doesn't because all three residues modulo 3 are represented.

It is conjectured that there are infinitely many
prime *k*-tuplets for each *k* (this would
be a simple consequence of Dickson's conjecture). However,
if this conjecture is true, it contradicts another well
known conjecture: that π(*x*+*y*) ≤ π(*x*)+π(*y*). (This
conjecture is a way of saying "primes thin out"). This second conjecture fails if we can find a *k*-tuple conjecture with length s(*k*)= 3159.

**See Also:** PrimeKtuple, PrimeKtupleConjecture, PrimeConstellation

**Related pages** (outside of this work)

- Prime k-tuplets (records, links, ...)
- K-Tuple Permissible Patterns

**References:**

- Forbes1999
T. Forbes, "Prime clusters and Cunningham chains,"Math. Comp.,68:228 (1999) 1739--1747.MR 99m:11007