Problem 825

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claims/: The 1 claim page of Problem 825, one per claimant's result; the problem's standing derives from them.

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Statement. Is there an absolute constant $C>0$ such that every integer $n$ with $\sigma(n)>Cn$ is the distinct sum of proper divisors of $n$?

Status. The site labels the problem PROVED (LEAN) (page last edited 1 February 2026). Larsen's proof and its Lean formalization are recorded on the claim page Larsen 2026.

Source. erdosproblems.com/825, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #825, https://www.erdosproblems.com/825.

References.

• [Gu04] Guy, Richard K., Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp.; B2 "Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers", printed p. 77: the weird-number questions, the last being "Can $\sigma(n)/n$ be arbitrarily large for weird $n$?", which Benkoski and Erdős conjecture has answer no, and for which Erdős offered a prize; a weird $n$ is abundant and not a sum of distinct proper divisors, so this question's constant $C$ is the conjectured bound on $\sigma(n)/n$. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures. Larsen's Lean proof is recorded on the claim page Larsen 2026.

Progress

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Known Results

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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.

• guy_2004_unsolved_problems_number_theory
• larsen_2026_sufficiently_abundant_numbers_pseudoperfect
• larsen_2026_sufficiently_abundant_numbers_pseudoperfect / corollary_5
• larsen_2026_sufficiently_abundant_numbers_pseudoperfect / theorem_1
• larsen_2026_sufficiently_abundant_numbers_pseudoperfect / theorem_4