
Glossary: Prime Pages: Top 5000: 
A prime ktuple 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 2tuples. Prime triplets are
3tuples. These have the patterns {p, p+2, p+6}
or {p, p+4, p+6}.
4tuples have the form {p, p+2, p+6, p+8}. There is a pair of twin primes in every prime 3tuple, and a prime 3tuple in every prime 4tuple (but not prime ktuple in every prime (k+1)tuple, 7tuples do not include 6tuples). So some authors use prime ktuplet to mean a prime ktuple 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 ktuple. 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 ktuplet is then defined as a sequence of consecutive primes {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 5tuplet , 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 ktuplets 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 (x+y) < (x)+(y). (This conjecture is a way of saying "primes thin out"). This second conjecture fails if we can find a ktuple conjecture with length s(k)= 3159.
See Also: PrimeKtuple, PrimeKtupleConjecture, PrimeConstellation Related pages (outside of this work)
References:
Chris K. Caldwell © 19992017 (all rights reserved)
