Erdős (Erdos) Problem #955: status in 2026

Erdős Problem #955

If A⊂ ℕ has density 0 then s^-1(A) must also have density 0. A conjecture of Erdős, Granville, Pomerance, and Spiro [EGPS90].

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #955 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_955 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #955 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_955`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 6 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Try it with your chatbot Submit a claim Follow (RSS)

Cite

@misc{cairn-erdos-955,
  title        = {Erdős Problem #955},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-955}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Ferdos-955.json)](https://cairn-commons.com/problems/erdos-955)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

If $A \subset N$ has density $0$ then $s^{- 1} (A)$ must also have density $0$ .

A conjecture of Erdős, Granville, Pomerance, and Spiro [EGPS90].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«955».

theorem erdos_955 :
    ∀ A : Set ℕ, A.HasDensity 0 → { x | s x ∈ A }.HasDensity 0

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/955. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/955
[EGPS90] Erdős, P. and Granville, A. and Pomerance, C. and Spiro, C., On the normal behavior of the iterates of some arithmetic functions. Analytic number theory (Allerton Park, IL, 1989) (1990), 165-204.
[Er73b] Erdős, P., Über die Zahlen der Form $σ (n) - n$ und $n - ϕ (n)$ . Elem. Math. (1973), 83--86.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[PPT18] Pollack, Paul and Pomerance, Carl and Thompson, Lola, Divisor-sum fibers. Mathematika (2018), 330--342.
[Po14b] Pollack, Paul, Some arithmetic properties of the sum of proper divisors and the sum of prime divisors. Illinois J. Math. (2014), 125--147.
[PoPo16] Pollack, Paul and Pomerance, Carl, Some problems of Erdős on the sum-of-divisors function. Trans. Amer. Math. Soc. Ser. B (2016), 1--26.
[Tr15] Troupe, Lee, On the number of prime factors of values of the sum-of-proper-divisors function. J. Number Theory (2015), 120--135.
[Tr20] Troupe, Lee, Divisor sums representable as the sum of two squares. Proc. Amer. Math. Soc. (2020), 4189--4202.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.