# P2958: Erdős's Conjecture on Divergent Reciprocal Sums and Arithmetic Progressions

- ID: `P2958`
- Reference: `erdos-problem-3`
- Page: https://theoremdb.org/statements/P2958
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 3: Let \(A\) be a set of natural numbers. If the sum of reciprocals \(\sum_{n \in A} \frac{1}{n}\) diverges, must \(A\) contain arbitrarily long arithmetic progressions? That is, does there exist, for every positive integer \(k\), a subset \(S \subseteq A\) that forms an arithmetic progression of length \(k\)?

### Context

This problem was posed by Paul Erdős and concerns the relationship between the analytic property of having a divergent sum of reciprocals and the combinatorial property of containing long arithmetic progressions. It connects additive combinatorics with the study of divergence of series.

### Problem setup

- **Definition (An arithmetic progression of length \(k\).** An arithmetic progression of length \(k\) is a set of the form \(\{a, a+d, a+2d, \ldots, a+(k-1)d\}\) for some integer \(a\) and positive integer \(d\).
- **Definition (A series \(\sum_{n \in A} \frac{1}{n}\).** A series \(\sum_{n \in A} \frac{1}{n}\) is said to diverge when the partial sums are not bounded, or equivalently, when the sequence of partial sums tends to infinity.
- **Definition (The notation \(\exists^\infty k\) in the formal statement).** The notation \(\exists^\infty k\) in the formal statement means 'there exist infinitely many \(k\)' or 'for arbitrarily large \(k\)'.
- **Remark.** This problem was posed by Paul Erdős and concerns the relationship between the analytic property of having a divergent sum of reciprocals and the combinatorial property of containing long arithmetic progressions. It connects additive combinatorics with the study of divergence of series.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 3: Let \(A\) be a set of natural numbers. If the sum of reciprocals \(\sum_{n \in A} \frac{1}{n}\) diverges, must \(A\) contain arbitrarily long arithmetic progressions? That is, does there exist, for every positive integer \(k\), a subset \(S \subseteq A\) that forms an arithmetic progression of length \(k\)?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 3 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 3: Let \(A\) be a set of natural numbers. If the sum of reciprocals \(\sum_{n \in A} \frac{1}{n}\) diverges, must \(A\) contain arbitrarily long arithmetic progressions? That is, does there exist, for every positive integer \(k\), a subset \(S \subseteq A\) that forms an arithmetic progression of length \(k\)? [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 3 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 3: Let \(A\) be a set of natural numbers. If the sum of reciprocals \(\sum_{n \in A} \frac{1}{n}\) diverges, must \(A\) contain arbitrarily long arithmetic progressions? That is, does there exist, for every positive integer \(k\), a subset \(S \subseteq A\) that forms an arithmetic progression of length \(k\)?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 3 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 3: Let \(A\) be a set of natural numbers. If the sum of reciprocals \(\sum_{n \in A} \frac{1}{n}\) diverges, must \(A\) contain arbitrarily long arithmetic progressions? That is, does there exist, for every positive integer \(k\), a subset \(S \subseteq A\) that forms an arithmetic progression of length \(k\)?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 3 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 3 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 3: Let \(A\) be a set of natural numbers. If the sum of reciprocals \(\sum_{n \in A} \frac{1}{n}\) diverges, must \(A\) contain arbitrarily long arithmetic progressions? That is, does there exist, for every positive integer \(k\), a subset \(S \subseteq A\) that forms an arithmetic progression of length \(k\)? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-3`, 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 3, maintained status record. Erdős Problems record 3, checked 2026-08-01. Problem 3; status field and linked bibliography https://www.erdosproblems.com/3
   - Also cited at Problem 3; 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's Conjecture on Divergent Reciprocal Sums and Arithmetic Progressions: 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 3 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's Conjecture on Divergent Reciprocal Sums and Arithmetic Progressions: 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 3. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/3.lean:L32; theorem erdos_3; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/3.lean#L32
   - 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's Conjecture on Divergent Reciprocal Sums and Arithmetic Progressions: Supplies the pinned formal declaration whose human-readable editorial statement is published here.