Claim: I actually have almost zero knowledge in this domain, and this answer is totally based on the correctness of this wiki page. Moreover, it could be that I misunderstood the content of that page, so it would be kind of some expert to point out to me if I made a mistake.

Quote from that wiki page:

More precisely, given any integer-valued polynomials $P_1,\cdots, P_k$ in one unknown $m$ all with constant term $0$, there are infinitely many integers $x$, $m$ such that $x + P_1(m), \cdots, x + P_k(m)$ are simultaneously prime.

From what I understand, this implies (or means) that there exists **positive** integers $x$ and $m$ such that all the $x + P_i(m)$ are primes.

Now for any fixed $k$, simply take $P_i(m) = m^i + im$. We will then have:
$$x + P_i(m) = (x + im) + m^i,$$
where $(x + im)_i$ and $(m^i)_i$ are arithmetic and geometric progressions, respectively.

atTerryTao. $\endgroup$1more comment