Erdős Problem #25 — open problem in number theory

Erdős Problem #25

Let n_1 < n_2 < … be an arbitrary sequence of integers, each with an associated residue class a_i pmodn_i. Let A be the set of integers n such that for every i either n < n_i or n not≡ a_i pmodn_i. Must the logarithmic density of A exist?

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-25,
  title        = {Erdős Problem #25},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-25}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $n_{1} < n_{2} < \dots$ be an arbitrary sequence of integers, each with an associated residue class $a_{i} (mod n_{i})$ . Let $A$ be the set of integers $n$ such that for every $i$ either $n < n_{i}$ or $n \neq \equiv a_{i} (mod n_{i})$ . Must the logarithmic density of $A$ exist?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«25». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_25 : answer(sorry) ↔
    ∀ (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ), (∀ i, 0 < seq_n i) → StrictMono seq_n →
      ∃ d, Set.HasLogDensity
        { x : ℕ | ∀ i, (x : ℤ) < seq_n i ∨ ¬((x : ℤ) ≡ seq_a i [ZMOD seq_n i]) } d

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/25. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/25

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.