The board / number theory / #692
Erdős Problem #692
DISPROVED LEANdatabase updated 2026-04-02
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The set of integers with exactly one divisor in the open interval \((n, m)\).
\(\delta_1(n,m)\) is the density of the set of integers with exactly one divisor in \((n,m)\).
The set of local maxima of the sequence `f` on `(a, ∞)`. Each local maximum is recorded by the first index `m` of a maximal interval `[m, b]` on which `f` is constant, with `f` strictly smaller at `m - 1` and at `b + 1`.
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)? Cambie has calculated that unimodularity fails even for \(n=2\) and \(n=3\). This was formalized in Lean by Monticone using Aristotle.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/692.Prizenone
Topicsnumber theory, divisors
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/692
Attempts
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