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Claim. The second question of Problem 1187 is answered no: there is a coloring of the integers with finitely many colors in which no monochromatic $k$-term arithmetic progression, $k\ge3$, has a prime common difference. Color each integer by its residue modulo $4$: two integers of one color differ by a multiple of $4$, which is never prime, so no color class even holds a pair with prime difference. With two colors, give the residues $0,1$ one color and $2,3$ the other: a $3$-term progression $a,a+p,a+2p$ with $p$ prime has $p$ odd, since a difference of $2$ joins the two classes, and then $a+2p\equiv a+2\pmod 4$ lies in the other class, so no monochromatic progression of three or more terms has a prime common difference. Both colorings are the site's, restated here in the words of this compilation.

Covers. The second question only: a finite coloring of the integers need not contain a monochromatic $k$-term progression with prime common difference. It says nothing about the first question, monochromatic progressions of primes, which Green and Tao's theorem answers yes on its accepted page.

Depends on. Nothing in this wiki.

Claimant and date. The argument is the commentary of the site's own problem page, unsigned and crediting nobody, whose recommended citation names the site's curator, T. F. Bloom, as the page's author; the page is filed under that name. The earliest dated revision in the page's history, of 8 April 2026, already carries the argument and gives the page its date; the current page was last edited the same day. Erdős's 1980 survey [Er80] poses both questions and answers neither.

Standing. Pending. The argument has no paper, preprint or forum discussion of its own, and the site's label is the claimant's own, since the curator who labels the problem wrote the argument, so it is not independent acceptance and no reviewed evidence is listed. Kenta Kitamura's Lean 4 development of 12 May 2026 formalizes the same counterexample for colorings of the natural numbers; it presents itself as a formalization of the standard counterexample described on the problem page and names no claimant, so it has its own pending page and gives this page no formalized evidence. The problem's standing is claimed through this page together with the accepted first-question claim.