---

Claim. T. H. Chan, Factors of a perfect square, Acta Arith. 163 (2014), no. 2, 141--143 (posted to arXiv as Perfect squares have at most five divisors close to its square root; card), Theorem 1.3: there is an absolute constant $C_0$ such that, for every $c\ge3$, every perfect square $n>e^{C_0c^6(\log c)^5}$ has at most five divisors in $[\sqrt n-cn^{1/4},\sqrt n+cn^{1/4}]$. Given $C>0$, take $c=\max(C,3)$: every large square has at most five divisors in $(n^{1/2},n^{1/2}+Cn^{1/4})$. So the answer to Problem 887 is yes for squares, with $K=5$, as the paper says of the Erdős-Rosenfeld question. The squares $(X_k-2)^2(X_k+2)^2$ built from $X^2-2Y^2=2$ have five divisors within $5n^{1/4}$ of $\sqrt n$, so five is optimal for the two-sided window.

Covers. The question with $n$ restricted to perfect squares.

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Arithmetica. The site labels the problem OPEN, so its commentary is not acceptance.