Problem 692

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claims/: The 1 claim page of Problem 692, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. Is $\delta_1(n,m)$ unimodular for $m>n+1$ (i.e. increases until some $m$ then decreases thereafter)? For fixed $n$, where does $\delta_1(n,m)$ achieve its maximum?

Statement (precise). Let $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. Is $\delta_1(n,m)$ unimodular for $m>n+1$ (i.e. increases until some $m$ then decreases thereafter)?

Notes. The site's wording prints two questions: whether $\delta_1(n,m)$ is unimodular in $m$, and, for fixed $n$, where it attains its maximum. Its label DISPROVED (LEAN) (page last edited 4 November 2025), which the site defines as solved in the negative, answers a yes-or-no question, and its commentary records only the answer to the first: Cambie's computations for $n=2$ and $n=3$ and Cambie's theorem [Ca25] that the sequence has superpolynomially many local maxima. The curator therefore reads the problem as the unimodality question, and the precise Statement keeps that question alone. Erdős's source supports the reading. In [Er79e], after the bound $\epsilon'(n,m)<c/(\log n)^\alpha$, Erdős writes: "Perhaps $\epsilon'(n,m)$ is unimodular for $m>n+1$, but I know nothing about this. I don't know where $\epsilon'(n,m)$ assumes its maximum." The first sentence is a conjecture; the second is a remark that the site rendered as a question. The change drops the second question from the Statement; nothing else changes. Under the full wording the problem is open: unimodality is disproved, while the maximizing $m$ is known only for $n=1$, where Cambie's Theorem 1 shows that $\delta_1(1,m)$ is non-increasing, so the maximum $1/2$ is attained at $m=3$ and $m=4$; for every $n\ge2$ no recorded source determines it, and that question is recorded as a variant with its own answer under Formulation. Under the precise Statement the problem is disproved, by Cambie's explicit dip $\delta_1(3,6)=7/20>\delta_1(3,7)=1/3<\delta_1(3,8)=38/105$, Cambie's computer check for $2\le n\le20$ and Theorem 3 of [Ca25], recorded as Cambie's claim page. The formal-conjectures file states both questions as parts and marks the maximizing part research open; it counts with the site's wording, not as a second ruling. The "(LEAN)" suffix rests on the Aristotle formalization of the finite example, linked from the claim page and not built here.

Formulation. The site's second question, where $\delta_1(n,m)$ attains its maximum for fixed $n$, is recorded here as a variant with its own answer. For $n=1$ it is answered by Cambie's Theorem 1: $\delta_1(1,m)$ is non-increasing in $m$, and $\delta_1(1,3)=\delta_1(1,4)=1/2>\delta_1(1,5)=1/3$, so the maximum $1/2$ is attained at $m=3$ and $m=4$. For every $n\ge2$ the variant is open: no recorded source determines the maximizing $m$, and the formal-conjectures file marks this part research open.

Status. DISPROVED (LEAN). The site's label describes the precise Statement, which Cambie's accepted claim disproves, so the derived standing is disproved. The maximizing-$m$ variant under Formulation is answered only for $n=1$ and does not bear on that standing. The Lean qualification refers to an Aristotle autoformalization of Cambie's finite example, linked from the claim page.

Source. erdosproblems.com/692, accessed 2026-09-05. Cite as: T. F. Bloom, Erdős Problem #692, https://www.erdosproblems.com/692, accessed 2026-09-05.

References.

• [Ca25] S. Cambie, Resolution of Erdős' problems about unimodularity. arXiv:2501.10333v1 (2025); Journal of Number Theory 280 (2026), 271--277, doi:10.1016/j.jnt.2025.08.014.
• [Er79e] Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82.
• [Fo08] Ford, Kevin, The distribution of integers with a divisor in a given interval. Annals of Mathematics (2) 168 (2008), 367--433.
• [Ob1] P. Erdős, Oberwolfach Mathematical Problems, Volume 1. Mathematisches Forschungsinstitut Oberwolfach (Various).

Formalization. Statement in formal-conjectures, as of 2026-08-08.

Current assessment

Cambie's explicit arithmetic example and Theorem 3 supply the mathematical disproof of unimodularity recorded on the claim page, which settles the precise Statement, so the problem's derived standing is disproved. Determining a maximizing $m$ for arbitrary fixed $n$ is the variant recorded under Formulation, answered only for $n=1$ and claimed by no one for $n\ge2$. Ford's Theorem 4, quoted under Known Results, is the analytic input to Cambie's proof. The formal-conjectures file has placeholders for its own proofs, and the Lean gist linked from the claim page, which this corpus has not built, does not supply the density bridge or a general UnimodularOn negation.

Progress

• Ca25 finite example: $\delta_1(3,6)=7/20>\delta_1(3,7)=1/3<\delta_1(3,8)=38/105$.
• Ca25, Theorem 1: $\delta_1(1,m)$ is non-increasing.
• Ca25, Claim 4 and Theorem 3: at the exponential scale, the sequence for large fixed $n$ has superpolynomially many local maxima.

Known Results

The interval is open: $(n,m)=\{n+1,\ldots,m-1\}$. The proved positive case $n=1$ in Cambie's paper contrasts with the strict dip at $n=3$ computed above. Cambie's Theorem 3 strengthens this to $\omega(\exp(n^c))$ local maxima for a fixed $c>0$ and all sufficiently large $n$. This proof does not determine the maximizing $m$ for an arbitrary fixed $n$, the variant recorded under Formulation.

The site's historical summary, with [Er79e] and [Fo08] as its references, reports the uniform estimate $\delta_1(n,m)\ll1/(\log n)^c$ for all $m$ and sharper Ford ranges. Ford's Theorem 4, quoted below, is the estimate Cambie's proof uses.

The analytic input is Ford's Theorem 4 (printed p. 375). With $H(x,y,z)$ counting integers up to $x$ with at least one divisor in $(y,z]$, and $H_1(x,y,z)$ counting those with exactly one, it gives

$$\frac{H_1(x,y,z)}{H(x,y,z)} \asymp_a \frac{\log\log(z/y+10)}{\log(z/y+10)}$$

for fixed $0<a<1$, sufficiently large $y$, $y+1\leq z\leq x^{5/8}$, and $yz\leq x^{1-a}$. Taking $y=n,z=m-1$ and then letting $x$ tend to infinity is the dependency used in Cambie's Claim 4.

The formal-conjectures file, as of 2026-08-08, has sorry placeholders for its own proofs, points erdos_692.parts.i through a formal_proof attribute to the Aristotle gist linked from the claim page, and labels the maximizing-$m$ part research open. The site's discussion thread links the pinned autoformalized Lean gist of 2 April 2026, which this corpus has not built. Its scope is periodicity, exact finite residue counts, and the strict dip for its rational residue-proportion definition; it does not bridge that ratio to HasDensity or prove a general UnimodularOn negation. The accepted mathematical disproof of unimodularity recorded on the claim page is Cambie's explicit arithmetic example and Theorem 3.

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.

• cambie_2025_resolution_erdos_problems_about_unimodularity
• cambie_2025_resolution_erdos_problems_about_unimodularity / claim_4
• cambie_2025_resolution_erdos_problems_about_unimodularity / example_3_6_8
• cambie_2025_resolution_erdos_problems_about_unimodularity / theorem_1
• cambie_2025_resolution_erdos_problems_about_unimodularity / theorem_3
• erdos_1979_unconventional_problems_number_theory_asterisque
• ford_2008_distribution_integers_divisor_given_interval
• ford_2008_distribution_integers_divisor_given_interval / theorem_4
• baker_2001_difference_between_consecutive_primes
• baker_2001_difference_between_consecutive_primes / theorem_1