Problem 1099

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claims/: The 2 claim pages of Problem 1099, one per claimant's result; the problem's standing derives from them.

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Statement. Let $1=d_1<\cdots<d_{\tau(n)}=n$ be the divisors of $n$, and for $\alpha>1$ let

$$h_\alpha(n) = \sum_i \left( \frac{d_{i+1}}{d_i}-1\right)^\alpha.$$

Is it true that

$$\liminf_{n\to \infty}h_\alpha(n) \ll_\alpha 1?$$

Status. Proved. The site credits Vose [Vo84]; the accepted claim is recorded on Vose's claim page, and Tenenbaum's 1987 theorem, which settles the question along the factorials, the least common multiples and the primorials, on its own claim page.

Source. erdosproblems.com/1099, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1099, https://www.erdosproblems.com/1099.

References.

• [Er81h] Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.
• [Te87] Tenenbaum, G., Sur un problème extrémal en arithmétique. Ann. Inst. Fourier (Grenoble) 37 (1987), no. 2, 1--18; doi:10.5802/aif.1083.
• [Vo84] Vose, Michael D., Integers with consecutive divisors in small ratio. J. Number Theory (1984), 233-238.

Formalization. Statement in formal-conjectures, pinned to the commit of 30 September 2026. The file, added on 20 September 2026, marks erdos_1099 solved with a formal_proof attribute pointing to Erdos1099.lean in Boris Alexeev's repository, a Lean file of 17 August 2026 that declares itself a formalization of a solution to the problem, with Vose as its informal author and Codex and GPT-5.6 Sol as its formal authors; it is linked on Vose's claim page. The same statement file marks its factorial and lcm variants solved, citing [Te87]. This corpus has not built either file, so no formalized evidence is listed.

Current assessment

The question is whether $\liminf_n h_\alpha(n)\ll_\alpha1$ for each fixed $\alpha>1$; the term $i=1$ makes the limit inferior at least $1$. Vose's 1984 construction answers yes, and Tenenbaum's 1987 Théorème 1 answers it again along the factorials, the least common multiples $\mathrm{lcm}(1,\dots,k)$ and the primorials, the sequences Erdős proposed, which the site's commentary (problem page last edited 19 October 2025) reports as unsettled. The standing derives from the two accepted claim pages: Vose's, on the curator's credit and the journal publication, and Tenenbaum's, on the journal publication alone. A thread comment of 4 July 2026 reports a chat transcript in which GPT-5.5 claims an improved bound, with no manuscript; it is not recorded as a claim. Neither proof has been reproduced or reviewed in this wiki. No dated literature search beyond the site's page and thread is recorded.

Known Results

• Vose's 1984 construction [Vo84]: $\liminf_n h_\alpha(n)\ll_\alpha1$ for every fixed $\alpha>1$; the site's accepted answer, on Vose's claim page.
• Tenenbaum's 1987 Théorème 1 [Te87]: $h_\alpha$ is bounded along $k!$, $\mathrm{lcm}(1,\dots,k)$ and the primorials, on Tenenbaum's 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.

• erdos_1981_problems_results_additive_multiplicative_number_theory
• erdos_1981_problems_results_additive_multiplicative_number_theory / display_1_1