Problem 449

Let $r(n)$ count the number of $d_1,d_2$ such that $d_1\mid n$ and $d_2\mid n$ and $d_1<d_2<2d_1$ . Is it true that, for every $\epsilon>0$ ,

r(n) < \epsilon \tau(n)

for almost all $n$ , where $\tau(n)$ is the number of divisors of $n$ ?

Source: erdosproblems.com/449

Number theory
Divisors

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

Disproved on the site: the curator credits Kevin Ford's observation that $r(n)>K\tau(n)$ holds on a set of positive density for every $K$ , deduced by a Cauchy-Schwarz bound from the dyadic divisor count of Problem 448, and cites Hall and Tenenbaum's book for the argument on an essentially identical problem; see the claim page (Ford, 2024). A December 2025 report in the discussion thread that ByteDance Seed's Seed-Prover 1.5 had solved this problem was judged by the curator a likely misnumbering and is not a claim. The standing in the frontmatter derives from the claim pages.

Problem 449

Status

Proof claims

Discussion