Problem 945

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Statement. Let $F(x)$ be the maximal $k$ such that there exist $n+1,\ldots,n+k\leq x$ with $\tau(n+1),\ldots,\tau(n+k)$ all distinct (where $\tau(m)$ counts the divisors of $m$). Estimate $F(x)$. In particular, is it true that

$$F(x) \leq (\log x)^{O(1)}?$$

In other words, is there a constant $C>0$ such that, for all large $x$, every interval $[x,x+(\log x)^C]$ contains two integers with the same number of divisors?

Status. Open.

Source. erdosproblems.com/945, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #945, https://www.erdosproblems.com/945.

References.

• [Er85e] Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.
• [ErMi52] Erdős, P. and Mirsky, L., The distribution of values of the divisor function $d(n)$. Proc. London Math. Soc. (3) (1952), 257-271.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0; section B18 "Solutions of $d(n)=d(n+1)$", printed p. 112, where the book reports the Erdős and Mirsky question with the guess $k=(\ln n)^c$. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Current assessment

The site labels the problem OPEN (page last edited 5 October 2025). Erdős and Mirsky [ErMi52] prove $(\log x)^{1/2}/\log\log x\ll F(x)\ll\exp(O((\log x)^{1/2}/\log\log x))$ (card); the upper bound comes from their count of the distinct values of $\tau$ up to $x$. A note by Adrian Beker, linked in the site's thread on 3 October 2025, lowers the upper bound to $\exp(O((\log x)^{1/3+o(1)}))$. Neither bound decides whether $F(x)\le(\log x)^{O(1)}$, so neither is a claim. The site also reports Cambie's observation that Cramér's conjecture, or a squarefree number in every interval of length $\gg\log x$ in $[x,2x]$, would give $F(x)\ll(\log x)^2$. That is a conditional yes to whether $F(x)\le(\log x)^{O(1)}$, but no manuscript of it is linked, so it has no claim page. Erdős [Er85e] said the lower bound could be raised to $(\log x)^{1-o(1)}$ but gave no proof.

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.

• erdos_1985_problems_results_number_theory
• erdos_1952_distribution_values_divisor_function
• erdos_1952_distribution_values_divisor_function / theorem_i
• erdos_1952_distribution_values_divisor_function / theorem_ii
• erdos_1952_distribution_values_divisor_function / theorem_v
• guy_2004_unsolved_problems_number_theory