Hardy-Littlewood prime tuples conjecture

The prime tuples conjecture asserts that for any fixed integers [math]\displaystyle{ a_1,\ldots,a_k }[/math], there exist infinitely many integers n such that [math]\displaystyle{ n+a_1,\ldots,n+a_k }[/math] are all prime, unless there is an "obvious" obstruction to this happening, namely that there exists some modulus q such that at least one of [math]\displaystyle{ n+a_1,\ldots,n+a_k }[/math] is not coprime to q for any n. There is a more quantitative version of this conjecture which predicts that the number of n of size N is roughly [math]\displaystyle{ {\mathfrak G} N / \log^k N }[/math], where the singular series [math]\displaystyle{ {\mathfrak G} }[/math] is a positive number depending on [math]\displaystyle{ a_1,\ldots,a_k }[/math] in an explicit fashion. For instance, for the question of finding twin primes n,n+2, the singular series is the twin prime constant

[math]\displaystyle{ {\mathfrak G} = 2 \prod_{p \hbox{odd}} 1 - \frac{1}{(p-1)^2}. }[/math]

Cramer's random model for the primes predicts this conjecture without the singular series factor, but by adding in the information that primes should almost always be coprime to any fixed modulus q, one can recover the prediction with the singular series factor.

The prime tuples conjecture also implies Poisson-type behaviour for the number of primes in [math]\displaystyle{ [n,n+\log n] }[/math] in the sense that any fixed moment of this number matches the Poisson prediction asymptotically (a calculation of Gallagher), although the accuracy here is not strong enough to say anything about Cramer's conjecture.