What is the source of the $k$-tuple conjecture, that every integer tuple $(k\_1,\ldots,k\_n)$ either contains all members of a congruence class mod a prime or has infinitely many primes amongst $(k\_1+c,\ldots,k\_n+c)\_{c\in\mathbb{N}}$? Of course there is also an expected density, so perhaps the forgoing is the weak form of the conjecture and the standard form gives that as well.


I've seen it attributed to Partitio Numerorum III several times, but I don't find it there. Conjecture B (p. 42) is the special case of pairs.


Schinzel's hypothesis H (in his joint paper with Sierpiński) generalizes this conjecture, but that was much later -- published in 1958.


Any ideas as to the source for either version of the conjecture? Or is it actually in Hardy-Littlewood and my reading skills have failed me?


* Hardy & Littlewood, "Some problems of 'partitio numerorum'; III -- On the expression of a number as a sum of primes".



Whence the k-tuple conjecture?
Quoting from "Linear Equations in Primes" by Green and Tao (see here for a preprint <http://arxiv.org/abs/math/0606088>):


"The name of Dickson is sometimes associated to this circle of ideas. In the
1904 paper [12], he noted the obvious necessary condition on the $a\_i$, $b\_i$ in order that the forms $(a\_1 n + b\_1,\dots, a\_t n + b\_t)$ might all be prime infinitely often and suggested that this condition might also be sufficient."


Where [12] is L. E. Dickson, A new extension of Dirichlet’s theorem on prime numbers, Messenger of Math. 33 (1904), 155–161.


There is more discussion on the history related to this type of problem in the paper of Green and Tao.


