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 which says essentially that for any $a_1,a_2 \cdots a_k$ and $b_1, b_2 \cdots b_k$, 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 which has been generalized to Hypothesis H, and the much stronger Bateman-Horn conjecture which makes a similar statement for polynomials of any degree, but also predicts asymptotics for how common the simultaneously prime values are.

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 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 which has been generalized to Hypothesis H, and the much stronger Bateman–Horn conjecture which makes a similar statement for polynomials of any degree, but also predicts asymptotics for how common the simultaneously prime values are.