Problem 693

../

---

Statement. Let $k\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\{a_1<a_2<\cdots \}$ be the set of those integers in $[n,n^k]$ which have a divisor in $(n,2n)$. Estimate

$$\max_{i} a_{i+1}-a_i.$$

Is this $\leq (\log n)^{O(1)}$?

Status. Open.

Source. erdosproblems.com/693, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #693, https://www.erdosproblems.com/693.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.