Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 690

claims/: The 2 claim pages of Problem 690, one per claimant's result; the problem's standing derives from them.

Statement. Let $d_k(p)$ be the density of those integers whose $k$ th smallest prime factor is $p$ (i.e. if $p_1<p_2<\cdots$ are the primes dividing $n$ then $p_k=p$ ).

For fixed $k\geq 1$ is $d_k(p)$ unimodular in $p$ ? That is, it first increases in $p$ until its maximum then decreases.

Formulation. The wording admits two readings, both raised in the site's thread: (i) whether $d_k(p)$ is unimodal for every fixed $k$ , a single yes-or-no question; (ii) for each fixed $k$ , whether $d_k(p)$ is unimodal. Erdős's source reads it as (i): he doubts that $d_v(p)$ is unimodal but has not disproved it ([Er79e], p. 75). The site's curator adopted (i) in the thread on 2026-05-07. The page's standing targets (i), which Cambie's theorem answers no. Under (ii), Cambie's theorem decides $k\le20$ ; every $k\ge21$ rests on the pending Wang–Crapis claim, which covers every $k\ge4$ .

Status. Solved; the site's label is SOLVED, decided in the thread on 2026-05-07 for Cambie's refereed theorem: unimodal for $k=1,2,3$ , not unimodal for $4\le k\le20$ , so the sequence is not unimodal for every fixed $k$ . The classification for every $k\ge4$ is the pending Wang–Crapis claim.

Source. erdosproblems.com/690, accessed 2026-09-05. Cite as: T. F. Bloom, Erdős Problem #690, https://www.erdosproblems.com/690, 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.

Formalization. Statement in formal-conjectures: erdos_690 answers False, with a sorry body whose formal_proof attribute, like those of its three variants hasDensity, cambie_unimodal and cambie_not_unimodal, points at the erdos_690 theorem of the Erdos690.lean file in Boris Alexeev's lean-proofs repository, at the pinned revision linked on Cambie's claim page, the file that declares itself a formalization of Cambie's result; the variant large_k, whether $d_k(p)$ is unimodal for some $k\ge21$ , is research open. The statement file is not itself a formalization; this corpus has not built or audited the linked file, so it gives no formalized evidence.

Current assessment

The compiled result is Cambie's Theorem 5 for $1\leq k\leq20$ , with exact rational recomputation of the finite checks recorded below. The claimed follow-up for every $k\geq4$ has a discussion endorsement and a source-local compilation whose symbolic route passed an independent review on 2026-09-07; its forty-two finite certificates are pending, so that compiled conclusion stays conditional and does not change this status. The two results are the problem's claim pages: Cambie's, accepted on the refereed publication and the site's decision of 2026-05-07, and Wang–Crapis's, claimed. Status search (2026-10-07): the site's page and the community database's entry for the problem, which records the status solved; the thread was accessed 2026-09-05; no literature database searched.

Progress

Ca25, Theorem 5: the sequence is unimodular for $k=1,2,3$ and has an explicit strict descent followed by a strict ascent for each $4\leq k\leq20$ .
Ca25, Claim 6: the exact recursion for the densities of integers divisible by exactly $r$ distinct primes among the first primes.

Known Results

Cambie's $d_k(p)$ counts the event that $p$ is the $k$ th smallest distinct prime divisor; exponents do not affect the event. His Theorem 5 proves unimodularity for $k=1,2,3$ and non-unimodularity for every $4\leq k\leq20$ . The finite checks in this transcription were recomputed with exact rational recurrence arithmetic. For example,

d_4(13)>d_4(17)<d_4(19),\qquad d_5(23)>d_5(29)<d_5(31).

The site's historical summary, attributed to [Er79e], reports a typical scale $e^{e^k}$ , a maximizing-prime scale $e^{(1+o(1))k}$ , and analogous non-unimodality for the kth-divisor sequence. Those original proofs are not compiled; the covered proof is [Ca25] Theorem 5 above.

The theorem does not claim the classification for every $k\geq4$ . A separate May 2026 preprint by Wang and Crapis, arXiv:2605.08542, claims non-unimodularity for every $k\geq4$ using a prime-gap threshold criterion, certified finite computations, and a uniform Chinese-remainder construction. The discussion endorsement by Nat Sothanaphan (12 May 2026) says that a standard check found no issues and that the proof could be regarded as correct, while noting a caveat about what the verifier presentation overclaims. That endorsement is follow-up progress recorded in the thread, not an independent review of this corpus's compilation; the site listed no proof claim for the problem on 2026-10-07. The all- $k$ claim is kept separate from Cambie's established finite result. Its source folder is Wang–Crapis 2026: Theorem 1.1 assembles the finite range, which reuses Cambie for $4\le k\le20$ and continues through $k=8600001$ , and the uniform CRT tail for $k\ge8600002$ . The symbolic route passed an independent review on 2026-09-07 (the filed record); its forty-two finite prime-enumeration, rational-sum and logarithmic certificates remain pending, and the analytic estimates, the constant $B$ enclosure and the two huge prime records are explicit imported premises, not newly compiled external proofs. The filing changes neither the status nor the compiled Cambie account above.

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.

Linked from (20)

Cambie's finite classification Wang–Crapis non-unimodality for every k at least 4 problems/arithmetic_functions/E0690/claims Arithmetic Functions Arithmetic Functions A Complete Answer to Erdős Problem 690 Independent reviews of the Wang–Crapis route Strict first differences of prime-factor densities Non-unimodality for every k at least four divisors/cambie_2025_resolution_erdos_problems_about_unimodularity Recurrence for the prime-divisor densities Unimodularity of the density of the $k$th prime factor divisors/erdos_1979_unconventional_problems_number_theory_asterisque Estimates of Some Functions Over Primes without R.H.Proposition 5.1 (p. 4): theta(x) - x < x/36260 for every x > 0 Proposition 6.6 (p. 8): p_k <= k(ln k + ln ln k - 1 + (ln ln k - 2)/ln k) for k >= 688383 Theorem 5.2 (p. 4): |theta(x) - x| < eta_k x / ln^k x for x >= x_k, with a table of constants Theorem 6.10 (p. 10): sum_{p<=x} 1/p - ln ln x - B lies within 1/(10 ln^2 x) + 4/(15 ln^3 x)Theorem 6.9 (p. 9): explicit lower and upper bounds for pi(x) in three forms, each with its own range New estimates for some functions defined over primes

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