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

Updated Oct 8

Problem 144

claims/: The 2 claim pages of Problem 144, one per claimant's result; the problem's standing derives from them.

Statement. The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is equal to $1$ .

Status. PROVED (LEAN).

Source. erdosproblems.com/144, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #144, https://www.erdosproblems.com/144.

References.

[Er64h] Erdős, P., On some applications of probability to analysis and number theory. J. London Math. Soc. (1964), 692-696.
[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
[ErHa79] Erdős, P. and Hall, R. R., The propinquity of divisors. Bull. London Math. Soc. (1979), 304-307.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section E3 "Density of integers with two comparable divisors", printed p. 313, which poses the question and reports the Maier and Tenenbaum solution. Library home: guy_2004_unsolved_problems_number_theory.
[MaTe84] Maier, H. and Tenenbaum, G., On the set of divisors of an integer. Invent. Math. (1984), 121-128.

Formalization. No statement in formal-conjectures; a Lean proof of the statement is linked from the claim page and described in the Current assessment.

Current assessment

The site's formulation (accessed 2026-09-04; the site's page was last edited 2026-04-08) states that the density of integers with two divisors $d_1<d_2<2d_1$ exists and equals one; Erdős also asked the form with $2$ replaced by any constant $c>1$ . Both hold, and the standing derives from one accepted full claim, Maier and Tenenbaum 1984, whose theorem gives almost all $n$ a pair of divisors with ratio below $1+(\log n)^{-\beta}$ for every $\beta<\log 3-1$ . The claim is refereed and accepted on the site curator's credit. The exponent is sharp by Erdős and Hall's 1979 paper, which also withdrew Erdős's earlier claim of the density-one statement; that claim, announced without proof in 1964 and restated in 1970, has its own page, Erdős 1964, with status withdrawn, and the Maier and Tenenbaum page records the sharpness. Guy's collection discusses the problem as E3.

The site's label carries a Lean qualification: the community database records that the resolution is formalized while no formal-conjectures statement file exists; the Lean proof lives in Boris Alexeev's repository and is linked from the claim page. The file is third-party Lean that this corpus has not built, so the claim lists no formalized evidence.

Search scope: the site's problem page, the community database (teorth/erdosproblems, data/problems.yaml), the formal-conjectures project and the Lean repository named above; no claim beyond the two pages was found. Nothing remains open in the stated question.

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 (19)

Erdős's withdrawn 1964 claim of two close divisors Maier and Tenenbaum, two close divisors for almost all integers problems/divisors/E0144/claims Divisors and Multiples Divisors and Multiples divisors/erdos_1964_applications_probability_analysis_number_theory Item 2 and displays (8), (9) (pp. 695--696): density of integers with two divisors in d_1 < d_2 < d_1(1 + (e/3)^{(1∓η) log log n})divisors/erdos_1970_extremal_problems_combinatorial_number_theory divisors/erdos_1978_unconventional_problems_divisors_integers divisors/erdos_1979_propinquity_divisors Theorem (p. 304, unnumbered): almost no n < x has divisors d < d' < d(1 + eta(x)(log d)^{1 - log 3})divisors/maier_1984_set_divisors_integer Theorem 1: Almost all integers have two very close divisors Theorem 2: A lower bound for the normal order of Hooley's Delta divisors/tenenbaum_2013_erdos_unconventional_problems_number_theory Statement (1) (pp. 2, 4): almost all n have divisors d_1 < d_2 < d_1(1 + (e/3)^{(1-eta) log log n}), which fails with 1 + eta; reported as a theorem Theorem 1 (p. 7): the limiting distribution nu of tau^+(n)/tau(n) is continuous at z = 1 number_theory/erdos_1979_unconventional_problems_number_theory_math_mag number_theory/guy_2004_unsolved_problems_number_theory

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