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Claim. If Hypothesis H holds then for every $k\ge3$ there are $k$ consecutive primes in arithmetic progression, and infinitely many such progressions, which would answer Problem 141 yes for every $k$. The source is A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), no. 3, 185--208. The paper states Hypothesis H: if finitely many irreducible polynomials with integer coefficients and positive leading coefficients have a product with no fixed prime divisor, then they take prime values simultaneously at infinitely many integers. Among the consequences it derives, $C_1$ (p. 190) is the existence of arbitrarily long arithmetic progressions of consecutive primes, and its sharpening $C_{1.4}$ (p. 191) states that for every $n>1$ and every $r$ divisible by all primes up to $n$ there are infinitely many progressions of $n$ consecutive primes with common difference $r$. The claim is conditional: Hypothesis H is unproven, so this page derives nothing for the problem's standing. Unconditionally the instances $k\le10$ are settled by computation on the Dubner et al. page.

Acceptance. Refereed: Acta Arithmetica, volume 4, issue 3 (1958), pp. 185--208; the Crossref record of the DOI gives these data. The site labels the problem OPEN (page last edited 28 September 2025) and its commentary does not cite this paper, so no reviewed evidence is listed. The corpus holds no card for the paper, has not checked the derivation and awards no tier of its own.

Depends on. Nothing in this wiki; the claim rests on the cited paper and on Hypothesis H.