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Let be the density of the set of integers with exactly one divisor in . Is unimodular for (i.e. increases until some then decreases thereafter)? For fixed , where does achieve its maximum?
Let be the density of the set of integers with exactly one divisor in . Is unimodular for (i.e. increases until some then decreases thereafter)?
Source: erdosproblems.com/692
An accepted solution exists. The statement is false.
DISPROVED (LEAN). The site's label describes the precise Statement, which Cambie's accepted claim disproves, so the derived standing is disproved. The maximizing- variant under Formulation is answered only for 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.
The site's wording prints two questions: whether is unimodular in , and, for fixed , 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 and 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 , Erdős writes: "Perhaps is unimodular for , but I know nothing about this. I don't know where 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 is known only for , where Cambie's Theorem 1 shows that is non-increasing, so the maximum is attained at and ; for every 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 , Cambie's computer check for and Theorem 3 of [Ca25], recorded as Cambie's claim page (2025). 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.