In one of his papers, Tao shows that set of Gaussian primes $\Bbb P[i]$ contains arbitrarily shaped constellations (where "shape" is any set of Gaussian integers and "constellation" of that shape is scaled and translated version of it).
I'm almost sure that I have seen the similar claim involving Cartestian product of set of (real) primes with itself, i.e. $\Bbb P^n$. However, I don't seem to be able to find any reference for that, and such reference is what I'm asking for. I might have also seen this as a conjecture somewhere.
Thanks in advance.
If you can scale and translate a collection $(a_1,b_1),\ldots,(a_k,b_k)$ to $(Aa_1+\alpha,Bb_1+\beta),\ldots,(Aa_k+\alpha,Bb_k+\beta)$, asking for all of the latter to be pairs of primes, then the $a_i$ and $b_i$ are entirely independent—it's no different from the one-dimensional problem. So I would credit this conjecture to Dickson.