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

- ID: `P2978`
- Reference: `erdos-problem-25`
- Page: https://theoremdb.org/statements/P2978
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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.

### Problem setup

- **Definition (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 (A sequence of positive integers).** A sequence of positive integers is strictly increasing if each term is strictly less than the next.
- **Definition (For integers \(a\), \(b\), and positive integer \(m\), we write \(a \equiv b \pmod{m}\) to mean that \(m\) divides \(a - b\).** For integers \(a\), \(b\), and positive integer \(m\), we write \(a \equiv b \pmod{m}\) to mean that \(m\) divides \(a - b\).
- **Remark.** 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.

### What 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?

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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?

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?

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-25`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 25, maintained status record. Erdős Problems record 25, checked 2026-08-01. Problem 25; status field and linked bibliography https://www.erdosproblems.com/25
   - 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 25 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 25. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/25.lean:L36; theorem erdos_25; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/25.lean#L36
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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.