---

Claim. The statement of Problem 144 holds, in a sharper form. In the survey P. Erdős, On some applications of probability to analysis and number theory, J. London Math. Soc. 39 (1964), 692--696 (carded at Erdős 1964), Erdős recalls that he had proved that the density of integers with two divisors $d_1<d_2<2d_1$ exists, and writes of the value one: "Unless I made a mistake I proved this recently" (p. 696). He states, in displays (8) and (9), that for every $\eta>0$ the integers $n$ with two divisors

$$d_1<d_2<d_1\bigl(1+(e/3)^{(1-\eta)\log\log n}\bigr)$$

have density one, while those with the exponent $1+\eta$ in place of $1-\eta$ have density zero, and adds that the proof of the first statement is comparatively simple and needs no probabilistic method. Since $(e/3)^{\log\log n}=(\log n)^{1-\log 3}$, the first statement gives almost all $n$ a pair of divisors with ratio below $1+(\log n)^{-\beta}$ for every $\beta<\log 3-1$, and in particular with $d_2<2d_1$, the problem. The paper gives no proof. Erdős restated the claim in 1970, for two coprime divisors $b_1<b_2<2b_1$, citing the 1964 paper and noting that "The proof has not been published and is quite complicated" (Some extremal problems in combinatorial number theory, 1970, p. 124; carded at Erdős 1970). The page is dated by the paper's year, 1964, as its first day.

Depends on. Nothing in this wiki.

Standing. Withdrawn. The introduction of P. Erdős and R. R. Hall, The propinquity of divisors, Bull. London Math. Soc. 11 (1979), 304--307 (carded at Erdős and Hall 1979), records of the density-one statement for $\beta<\log 3-1$ that "this claim has had to be withdrawn" (p. 304); the same paper proves the density-zero half in a quantitative form. No proof of the 1964 claim was published. The site's discussion thread brought the 1964 and 1970 statements and the withdrawal together in comments of 27 November 2025, after which the site added the Erdős--Hall paper to its references. The statement itself was proved in 1984 by Maier and Tenenbaum, whose accepted claim is on their page; the problem's standing derives from that page alone.