# P3046: Consecutive Primes in Arithmetic Progression

- ID: `P3046`
- Reference: `erdos-problem-141`
- Page: https://theoremdb.org/statements/P3046
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?

### Context

This problem concerns the existence of arbitrarily long sequences of consecutive primes that form arithmetic progressions. The question asks whether for every length \(k \geq 3\), there exists some starting prime such that the \(k\) consecutive primes beginning at that point are equally spaced.

### Problem setup

- **Definition (A set \(s\) of natural numbers).** A set \(s\) of natural numbers is an arithmetic progression of length \(l\) if there exist natural numbers \(a\) (the first term) and \(d\) (the common difference) such that \(s = \{a + nd \mid n \in \mathbb{N}, n < l\}\).
- **Definition (A set \(s\) of natural numbers).** A set \(s\) of natural numbers is a prime progression of length \(l\) if there exists a natural number \(a\) such that \(s\) has cardinality \(l\) and equals the set of the first \(l\) consecutive primes starting from the \(a\)-th prime, i.e., \(s = \{p_{a+n} \mid n \in \mathbb{N}, n < l\}\) where \(p_m\) denotes the \(m\)-th prime number.
- **Definition (A set \(s\).** A set \(s\) is an arithmetic progression of consecutive primes of length \(k\) if it is both an arithmetic progression of length \(k\) and a prime progression of length \(k\).
- **Remark.** This problem concerns the existence of arbitrarily long sequences of consecutive primes that form arithmetic progressions. The question asks whether for every length \(k \geq 3\), there exists some starting prime such that the \(k\) consecutive primes beginning at that point are equally spaced.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 141 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes? [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 141 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 141 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 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 141 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 141 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 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-141`, 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 141, maintained status record. Erdős Problems record 141, checked 2026-08-01. Problem 141; status field and linked bibliography https://www.erdosproblems.com/141
   - Also cited at Problem 141; 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 Consecutive Primes in Arithmetic Progression: 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 141 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 Consecutive Primes in Arithmetic Progression: 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 141. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/141.lean:L75; theorem erdos_141; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/141.lean#L75
   - 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 Consecutive Primes in Arithmetic Progression: Supplies the pinned formal declaration whose human-readable editorial statement is published here.