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Claim. The Main Theorem of K. Ramachandra, T. N. Shorey and R. Tijdeman, On Grimm's problem relating to factorisation of a block of consecutive integers, J. Reine Angew. Math. 273 (1975), 109--124: there is an effectively computable absolute constant $\alpha_3>0$ such that, for $n\geq3$ and $g=[\alpha_3(\log n)^3(\log\log n)^{-3}]$, there are pairwise distinct primes $p_1,\ldots,p_g$ with $p_i\mid n+i$ for $i=1,\ldots,g$. The paper was received on 1972-10-28; the page is dated by the journal issue.

Covers. Every run of composites $n+1,\ldots,n+k$ with $k\leq\alpha_3(\log n/\log\log n)^3$, for which the problem's distinct primes exist; this answers the question of Problem 375 for those runs. The constant $\alpha_3$ is very small, so the range is empty for small $n$, and longer runs are not covered. The formal-conjectures statement file records this range as a variant of the problem.

Acceptance. The result appeared in a refereed journal, the Journal für die reine und angewandte Mathematik, in 1975: the refereed evidence. The site labels the problem FALSIFIABLE, an open label, so its commentary's credit is not reviewed evidence. The range of Grimm's theorem lies inside this one for all large $n$.