Problem 886

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claims/: The 1 claim page of Problem 886, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilon(1)$?

Status. The site labels the problem OPEN. The answer is yes for every $\epsilon\ge1/4$ by Erdős and Rosenfeld's bound; the range $0<\epsilon<1/4$ is open.

Source. erdosproblems.com/886, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #886, https://www.erdosproblems.com/886.

References.

• [ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353-359.

Formalization. Statement in formal-conjectures.

Current assessment

The site labels the problem OPEN (page last edited 1 February 2026). For every $\epsilon\ge1/4$ the answer is yes: the window lies within $n^{1/4}$ of $\sqrt n$, where the bound of Erdős and Rosenfeld allows at most two divisors for every $n$ (claim page). For $0<\epsilon<1/4$ the problem is open. Their Proposition 4.2 gives infinitely many $n$ with four divisors within about $8n^{1/4}$ of $\sqrt n$ (the site prints $16n^{1/4}$, the paper's bound on the factor differences), which settles no instance. Letendre's preprint (card), cited in the site's thread, gives $O(1/(\epsilon-1/4))$ divisors for every $\epsilon>1/4$ (its Proposition 1 at $\theta=1/2$). That range is already settled with a stronger bound, and the preprint is not refereed, so it has no claim page.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• chan_2015_factors_almost_squares_lattice_points_circles
• chan_2015_factors_almost_squares_lattice_points_circles / theorem_2
• erdos_1997_factor_difference_set_integers
• letendre_2025_divisors_integer_short_interval
• letendre_2025_divisors_integer_short_interval / conjecture_1
• letendre_2025_divisors_integer_short_interval / proposition_1
• letendre_2025_divisors_integer_short_interval / theorem_1
• letendre_2025_divisors_integer_short_interval / theorem_2