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Level A · Machine-checkable Hard Number theory P-erdos-955

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].

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.

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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 2026-09-28}
}

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Claims
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Verified
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Disputed
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Refuted
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

If has density then must also have density .

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«955». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_955 :
    answer(sorry) ↔
      ∀ 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 und . 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.
  • [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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.