Let S be a finite set of integers. Then S is the pattern for a set of prime numbers P if there is some n such that P = { n + s | s ∈ S}.

S is admissible if it doesn't intersect all of the residue classes mod p for any prime p. The prime patterns conjecture states that any finite, admissible set is the pattern for infinitely many sets of primes. In other words, for any admissible S, there are infinitely many values of n for which n + s₁, n + s₂, ..., n + sₖ are all prime (where s₁, s₂, ..., sₖ are the elements of S).

This is a generalization of all of the following conjectures:

/JosephNoonan/is-de-polignacs-conjecture-true

/NcyRocks/are-there-infinitely-many-twin-prim

/JosephNoonan/are-there-infinitely-many-cousin-pr

/NcyRocks/are-there-infinitely-many-sexy-prim

/JosephNoonan/are-there-infinitely-many-prime-tri