Let d_k(p) be the density of those integers whose kth smallest prime factor is p (i.e. if p_1<p_2<… are the primes dividing n then p_k=p). For fixed k≥ 1 is d_k(p) unimodular in p? That is, it first increases in p until its maximum then decreases.

Erdős Problem #690 — Let d_k(p) be the density of those integers whose kth smallest prime factor is p (i.

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.

Erdős asked whether, for each fixed $k$ , the densities indexed by the prime $p$ rise to one maximum and then fall. The known classification is complete for $1\leq k\leq 20$ , while the remaining cases had been open.

Known results

Cambie proved unimodality for $1\leq k\leq 3$ .
Cambie proved non-unimodality for $4\leq k\leq 20$ .
Cambie proved eventual decrease for every fixed $k$ .

Recent claimed complete resolution

Shouqiao Wang and Davide Crapis claim that the sequence is non-unimodal for every $k\geq 4$ , using prime-gap arguments, certified computations, and a Chinese-remainder construction. Their preprint says the proof was produced by “Fields” with limited human interaction; this is an unverified claim, with no independent expert assessment found.

Community submission (unverified), October 28, 2026

A submitted paper argues that non-unimodality holds for every $k\geq 4$ , which would settle the classification; the submission itself is unverified.

Current status (as of October 2026): The cases $1\leq k\leq 3$ and $4\leq k\leq 20$ are established, while the claimed result for every $k\geq 4$ remains unverified.

19 declaration(s) are outside `namespace Submission` (Nat.count_mul_period, Nat.count_eq_mod_add_div_mul, Nat.countSubtypeFinEquiv). Everything an answer adds goes inside it, so two answers can never collide

Erdős Problem #690 — Let $d_k(p)$ be the density of those integers whose $k$ th smallest prime factor is $p$ (i.

Progress summary

Solutions 1