As written, this is hopeless false. $(2,3)$ is an obvious counterexample. Slightly less trivially, $(3,5,7)$ is a counterexample. 

One can correct for these, and if one does so, one gets a version of the [Dickson conjecture][1] which says essentially that for any $a_1,a_2 \dotsc a_k$ and $b_1, b_2 \dotsc b_k$, there are infinitely many $n$ where $a_i + b_i n$ are prime for all $1 \leq i \leq k$ unless there is an obvious divisibility restriction preventing this from happening. See also the Hardy–Littlewood $k$-tuples conjecture.

See also the this is the [Bunyakovsky conjecture][2] 
which has been generalized to [Hypothesis H][3], and the much stronger [Bateman–Horn conjecture](https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture) which makes a similar statement for polynomials of any degree, but *also* predicts asymptotics for how common the simultaneously prime values are.


  [1]: https://en.wikipedia.org/wiki/Dickson%27s_conjecture
  [2]: https://en.wikipedia.org/wiki/Bunyakovsky_conjecture
  [3]: https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H