slug: erdos-1979-exactly-one-divisor-unimodality bibkey: tenenbaum2013unconventional doi: 10.1007/978-3-642-39286-3_23 triage: theorem motivation_gids:

• D5/S3/Arith/Density/ErdosExactlyOneDivisorUnimodalityRefutation

Refutation of the one-divisor interval-density unimodality suggestion

Problem

Erdős, Some unconventional problems in number theory, Astérisque 61 (1979), printed page 78:

Perhaps ε₁(n,m) is unimodular for m > n+1, but I know nothing about this.

Here ε₁(n,m) is the natural density of integers N having precisely one divisor d with n < d < m. Tenenbaum, Some of Erdős’ Unconventional Problems in Number Theory, Thirty-four Years Later (2013), printed page 18, states:

To my knowledge, the question of unimodality of ε₁(y,z) as a function of z is still open.

Tenenbaum writes the upper endpoint as d <= z; the formal refutation follows the strict upper endpoint in Erdős’s original sentence.

Motivation

The 1979 paper poses an elementary distribution-of-divisors question, and the 2013 chapter records it as still open. Exact densities for one fixed value of n can refute the unqualified assertion without an asymptotic estimate.

Gap

The question is erdosproblems.com problem 692 (https://www.erdosproblems.com/692), recorded there as disproved: Stijn Cambie, Resolution of Erdős’ problems about unimodularity, arXiv:2501.10333 (2025), shows that δ₁(n,m) is not unimodal for n = 2 and n = 3 and that for fixed n the sequence has superpolynomially many local maxima; a Lean formalization of the negative answer by Monticone is linked from that page, and google-deepmind/formal-conjectures (FormalConjectures/ErdosProblems/692.lean) marks the unimodality part solved with the same reference. The refutation below therefore certifies, in this repository’s kernel, a statement already refuted in 2025; it is not a first resolution. Tenenbaum’s 2013 remark predates Cambie’s paper. R. R. Hall’s 1992 paper On some conjectures of Erdős in Astérisque, I, DOI 10.1016/0022-314X(92)90096-8, remains unread here (publisher HTTP 403).

Route

For fixed n,m, membership in the set of integers having exactly one divisor in (n,m) is periodic with period the least common multiple of the integers in that interval. Its density is obtained by binding the frozen theorem D5/S3/ObserverMemory/Prediction/EventualCycleAverage.eventual_cycle_average to the finite orbit on Fin P with the event’s zero-one indicator as the observable. The resulting cycle average is the number of members in [0,P) divided by P.

For n=2, the periods and favorable residue counts at m=6,7,8 are (60,26), (60,22), and (420,156). Hence the respective densities are 13/30, 11/30, and 13/35, a strict valley. Weak unimodality means that there is a mode at or after the beginning of the tail, with the function nondecreasing up to the mode and nonincreasing from the mode onward. The theorem not_unimodal_of_valley proves that any strict valley on the tail rules out this shape: a mode after the valley bottom contradicts the preceding fall, while a mode at or before it contradicts the later rise.

Falsifier

A correction to any of the three period counts, a failure of the periodic-set density theorem, or evidence that the printed assertion had an omitted lower bound on n would invalidate this refutation of the stated claim.

Evidence

• Module: D5/S3/Arith/Density/ErdosExactlyOneDivisorUnimodalityRefutation.lean.
• Frozen density input: D5/S3/ObserverMemory/Prediction/EventualCycleAverage.eventual_cycle_average.
• Escape content: not_unimodal_of_valley.
• Formal claim and refutation: claim and result.
• Axiom closure for both not_unimodal_of_valley and result: std3 (propext, Classical.choice, Quot.sound).
• Formal exact values: ε₁(2,6)=13/30, ε₁(2,7)=11/30, and ε₁(2,8)=13/35, from full periods with counts 26/60, 22/60, and 156/420.
• The orchestrator independently enumerated the six full-period values for m=4,...,9: 1/3, 5/12, 13/30, 11/30, 13/35, 11/35, with periods 3, 12, 60, 60, 420, 840 and favorable counts 1, 5, 26, 22, 156, 264.

Triage

theorem. The universal printed suggestion is refuted by the n=2 strict valley, and the resolution is attached only to result.

ASSUMED-UNVERIFIED

Hall’s 1992 paper was not read. The literature search is bounded to the checked surfaces listed above and does not establish first-publication priority. Only the unqualified printed statement is refuted; a repaired variant restricted to sufficiently large n is not addressed. The separate question of where the density attains its maximum is not addressed. GitHub code search required an authenticated session and was not checked.