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on Erdős #1212 · number theory / primes

The statement

Let $G$ be the graph with vertex set those pairs $(x,y)\in \mathbb{N}^2$ with $\mathrm{gcd}(x,y)=1$ , in which we join two vertices if the differ in only one coordinate, and there by $\pm 1$ .

Is there a path going to infinity on $G$ , say $P$ , such that for all $(x,y)\in P$ both $\min(x,y)>1$ and at least one of $x$ or $y$ is composite?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.