Claimed. The site labels the problem SOLVED, but the derived standing is claimed: the first question's yes is an accepted claim, while the second question's no rests only on the site's own modulo-4 argument and Kitamura's unbuilt Lean proof of it for the natural numbers, both pending claim pages, since the argument has no publication and the curator who labels the problem wrote it. The label is the site's (SOLVED, page last edited 8 April 2026, as of 2026-10-07). The two questions have different answers: the first is yes for every $k\ge3$ by the Green–Tao theorem [GrTa08], since some color class of a finite coloring has positive relative upper density in the primes and so contains $k$-term progressions; the second is no, by giving each integer the color of its residue class modulo $4$, under which two integers of one color differ by a multiple of $4$, or with two colors by putting the residues $0,1$ in one class and $2,3$ in the other, which has no monochromatic $3$-term progression with prime difference.