Cambie 2025 resolution erdos problems about unimodularity

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claim_4: Establishes the comparison between zero and one divisor densities used to create many local maxima in problem 692.

claim_6: Gives the exact recursion for the density of integers divisible by a fixed number of the first distinct primes.

example_3_6_8: Gives the exact three-term computation that disproves unimodality for the divisor interval problem at n equal to 3.

theorem_1: Proves that the one-divisor density for intervals beginning at 1 is non-increasing, the exceptional positive case in problem 692.

theorem_3: Uses Ford's divisor-interval estimate and prime gaps to produce many alternating rises and falls in the one-divisor density.

theorem_5: Proves unimodularity for k equal to 1, 2, or 3 and gives exact finite valleys proving non-unimodality for 4 through 20.

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Stijn Cambie, Resolution of Erdős' problems about unimodularity. arXiv:2501.10333v1 (17 January 2025). The peer-reviewed version is in Journal of Number Theory 280 (2026), 271--277, https://doi.org/10.1016/j.jnt.2025.08.014; the retained PDF here is the open arXiv v1. The arXiv record (https://arxiv.org/abs/2501.10333, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Writing delta_1(n,m) for the density of integers with exactly one divisor in the open interval (n,m), the paper answers Erdős problem 692 in the negative: computation shows the sequence is not unimodal for 2 <= n <= 20 (an explicit hand check gives delta_1(3,6) = 7/20, delta_1(3,7) = 1/3, delta_1(3,8) = 38/105, so the sequence dips), and Theorem 3 shows that for some c > 0 the sequence (delta_1(n,m))_m has omega(exp(n^c)) local maxima, so it is very far from unimodal. Theorem 1 proves the positive case n = 1: delta_1(1,m) is non-increasing in m, hence unimodular, proved by an induction over m in which L = prod p_i^2 prod q_i and A, the count of residues with exactly one divisor in {2,...,m-1}, satisfy A/L non-increasing and A >= phi(L). For problem 690, Theorem 5 shows the density d_k(p) of integers whose kth smallest prime factor is p is unimodular for k in {1,2,3} but not unimodular for every 4 <= k <= 20, via a recursion for the densities delta_r(i) of integers with exactly r distinct prime divisors among the first i+1 primes p_0 = 2, ..., p_i. The paper therefore gives the proved boundary cases in this finite range; a broader every-k >= 4 claim appears in a May 2026 preprint and is recorded separately on the problem page. The discussion records a named 12 May 2026 standard-check endorsement with a caveat about verifier presentation; this source folder does not compile that follow-up manuscript.

Source: https://arxiv.org/abs/2501.10333.

Bears on. #690, #692

Results to transcribe.

• Theorem 1: delta_1(1,m) is non-increasing in m and hence unimodular.
• Claim 4: for some c > 0, all large n and m = Theta(exp(3 n^c)), delta_0(n,m+1) > delta_1(n,m+1), the engine of the many-local-maxima proof.
• Theorem 3: for some c > 0, (delta_1(n,m))_{m >= n+2} has omega(exp(n^c)) local maxima.
• Claim 6: recursion for delta_r(i), the density with exactly r distinct prime divisors among the first i+1 primes p_0 = 2, ..., p_i.
• Theorem 5: d_k(p) is unimodular for k in {1,2,3} and not unimodular for every 4 <= k <= 20.
• Finite example: delta_1(3,6) = 7/20 > delta_1(3,7) = 1/3 < delta_1(3,8) = 38/105.