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 b₁ < b₂ < ... < b_k, with b_k - b₁ = s and, for every prime q, not all the residues modulo q are represented by b₁, b₂, ..., b_k. A prime k-tuplet is then defined as a sequence of consecutive primes {p₁, p₂, ..., p_k}such that for every prime q, not all the residues modulo q are represented by p₁, p₂, ..., p_k, and p_k - p₁ = 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.

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