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 $(a_1 n + b1,\dots, a_t n + b_t)$ 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.

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.