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Claim. There is an $n_0$ such that for every $n\geq n_0$, with $g=[\log n/(2\log\log n)]$, there are pairwise distinct primes $p_1,\ldots,p_g$ with $p_i\mid n+i$ for $1\leq i\leq g$. The result is in C. A. Grimm, A conjecture on consecutive composite numbers, Amer. Math. Monthly 76 (1969), no. 10, 1126--1128, the paper that states the conjecture of Problem 375; the statement above is the form in which Ramachandra, Shorey and Tijdeman report Grimm's theorem in the introduction of their 1975 paper (claim page). The journal record gives the December 1969 issue and no day, so the page is dated by the first of that month.

Covers. Every run of composites $n+1,\ldots,n+k$ with $n\geq n_0$ and $k\leq\log n/(2\log\log n)$, for which the problem's distinct primes exist. Longer runs are not covered, so the problem stays open.

Acceptance. The result appeared in a refereed journal, the American Mathematical Monthly, in 1969: the refereed evidence. The site labels the problem FALSIFIABLE, an open label, so its commentary's credit is not reviewed evidence.