[#P2978] Erdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance Sets

Problem. Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Define \(A\) to be the set of natural numbers \(x\) such that for every index \(i\), either \(x < n_i\) or \(x \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist?

A finite mathematical diagram showing residue classes removed from nested integer moduli. — Residue classes marked on several modular cycles.

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Definitions and notation

1Context

This problem originates from Erdős and concerns the existence of logarithmic density for sets defined by avoiding residue classes modulo a growing sequence of moduli, where the avoidance condition only applies once the modulus exceeds the candidate integer.

2Problem setup

Definition 1 (The logarithmic density of a set \(S \subseteq \mathbb{N}\). The logarithmic density of a set \(S \subseteq \mathbb{N}\) is said to exist with value \(d\) if the limit \(\lim_{N \to \infty} \frac{1}{\log N} \sum_{\substack{x \in S \\ x \leq N}} \frac{1}{x}\) exists and equals \(d\), where the sum is understood to be \(0\) when \(N = 1\).

Definition 2 (A sequence of positive integers). A sequence of positive integers is strictly increasing if each term is strictly less than the next.

Remark 1. For integers \(a\), \(b\), and positive integer \(m\), we write \(a \equiv b \pmod{m}\) to mean that \(m\) divides \(a - b\).

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Define \(A\) to be the set of natural numbers \(x\) such that for every index \(i\), either \(x < n_i\) or \(x \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist?

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 25 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Define \(A\) to be the set of natural numbers \(x\) such that for every index \(i\), either \(x < n_i\) or \(x \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist?^[1]

1Packet records

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 25 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Define \(A\) to be the set of natural numbers \(x\) such that for every index \(i\), either \(x < n_i\) or \(x \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist?	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 25 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Define \(A\) to be the set of natural numbers \(x\) such that for every index \(i\), either \(x < n_i\) or \(x \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist?^[1] Relevance to this problem For erdos problem 25, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 25 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 25 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Define \(A\) to be the set of natural numbers \(x\) such that for every index \(i\), either \(x < n_i\) or \(x \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Define \(A\) to be the set of natural numbers \(x\) such that for every index \(i\), either \(x < n_i\) or \(x \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist?	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Define \(A\) to be the set of natural numbers \(x\) such that for every index \(i\), either \(x < n_i\) or \(x \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist?^[1] Relevance to this problem For Erdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance Sets, record erdos-problem-25-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 25: Let \(n_1 < n_2 < \dots\) be an arbitrary strictly increasing sequence of positive integers, and for each \(i\) let \(a_i\) be an integer residue. Define \(A\) to be the set of natural numbers \(x\) such that for every index \(i\), either \(x < n_i\) or \(x \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 25 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 25 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemErdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance Sets

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Erdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance Sets.” TheoremDB. P2978. Problem statement; statement text SHA-256 d32b923c620bd8be4425bc8b0eafc12eee360c47d9d1ebf34a83c1ecf4d79397. https://theoremdb.org/statement/?ref=P2978

BibTeX

@misc{theoremdb-problem-d32b923c620bd8be4425bc8b0eafc12eee360c47d9d1ebf34a83c1ecf4d79397,
  title = {{Erdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance Sets}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 d32b923c620bd8be4425bc8b0eafc12eee360c47d9d1ebf34a83c1ecf4d79397},
  url = {https://theoremdb.org/statement/?ref=P2978}
}

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This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Erdős Problems database, Problem 25, maintained status record. Erdős Problems record 25, checked 2026-08-01. Problem 25; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 25; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Erdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance Sets: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 25 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Erdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance Sets: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 25. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/25.lean:L36; theorem erdos_25; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Erdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance Sets: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

[#P2978] Erdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance Sets

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