Problem 486

Let $A\subseteq \mathbb{N}$ , and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$ . Let

B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\textrm{ with }m>n\}.

Must $B$ have a logarithmic density, i.e. is it true that

\lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ m<x}}\frac{1}{m}

exists?

Source: erdosproblems.com/486

Number theory
Primitive sets

Wiki page

Status

ClaimedDisproved

A full solution has been claimed but not yet accepted. The statement is false.

OPEN, the site's label (2026-10-06). The derived standing departs from it because a 2026 manuscript of Wang, produced with GPT-5.6 Sol, claims a disproof by a fixed delayed congruence system whose survivor set has no logarithmic density; it is a pending full claim on its claim page (Wang, 2026), which makes the problem claimed as disproved, and it has no refereed publication, site acceptance or review by a named reviewer. The case $X_n=\{0\}$ , which the site's remark credits to Davenport and Erdős, is an accepted partial claim on its claim page (Davenport Erdos, 1936).

Be34: Besicovitch, A., On the density of certain sequences of integers. Math. Annalen (1934), 336-341.
DaEr36: Davenport, H. and Erdős, P., On sequences of positive integers. Acta Arithmetica (1936), 147-151.
DaEr51: Davenport, H. and Erdős, P., On sequences of positive integers. J. Indian Math. Soc. (N.S.) (1951), 19-24.
Er80: Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

Problem 486

Status

Proof claims

Discussion