# P2972: Infinitude of Cluster Primes

- ID: `P2972`
- Reference: `erdos-problem-17`
- Page: https://theoremdb.org/statements/P2972
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 17: A prime number $p$ is called a cluster prime if every even natural number $n \le p - 3$ can be written as a difference of two primes $q_1 - q_2$ with $q_1, q_2 \le p$. Are there infinitely many cluster primes?

### Context

This is Problem 17 from the collection of Erdős problems. The counting function of cluster primes up to $x$, denoted $\pi^{\mathcal{C}}(x)$, satisfies strong upper bounds: Blecksmith, Erdős, and Selfridge (1999) proved $\pi^{\mathcal{C}}(x) \ll_A x(\log x)^{-A}$ for every $A > 0$, and Elsholtz (2003) refined this to $\pi^{\mathcal{C}}(x) \ll x\exp(-c(\log\log x)^2)$ for every $0 < c < 1/8$. It is known that $97$ is the smallest prime that is not a cluster prime.

### Problem setup

- **Definition (A natural number $p$).** A natural number $p$ is a cluster prime if $p$ is prime and for every even natural number $n$ with $n \le p - 3$, there exist primes $q_1$ and $q_2$ such that $q_1 \le p$, $q_2 \le p$, and $n = q_1 - q_2$.
- **Remark.** This is Problem 17 from the collection of Erdős problems. The counting function of cluster primes up to $x$, denoted $\pi^{\mathcal{C}}(x)$, satisfies strong upper bounds: Blecksmith, Erdős, and Selfridge (1999) proved $\pi^{\mathcal{C}}(x) \ll_A x(\log x)^{-A}$ for every $A > 0$, and Elsholtz (2003) refined this to $\pi^{\mathcal{C}}(x) \ll x\exp(-c(\log\log x)^2)$ for every $0 < c < 1/8$. It is known that $97$ is the smallest prime that is not a cluster prime.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 17: A prime number $p$ is called a cluster prime if every even natural number $n \le p - 3$ can be written as a difference of two primes $q_1 - q_2$ with $q_1, q_2 \le p$. Are there infinitely many cluster primes?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 17 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 17: A prime number $p$ is called a cluster prime if every even natural number $n \le p - 3$ can be written as a difference of two primes $q_1 - q_2$ with $q_1, q_2 \le p$. Are there infinitely many cluster 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 17 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 17: A prime number $p$ is called a cluster prime if every even natural number $n \le p - 3$ can be written as a difference of two primes $q_1 - q_2$ with $q_1, q_2 \le p$. Are there infinitely many cluster primes?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 17 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 17: A prime number $p$ is called a cluster prime if every even natural number $n \le p - 3$ can be written as a difference of two primes $q_1 - q_2$ with $q_1, q_2 \le p$. Are there infinitely many cluster primes?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 17 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 17 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 17: A prime number $p$ is called a cluster prime if every even natural number $n \le p - 3$ can be written as a difference of two primes $q_1 - q_2$ with $q_1, q_2 \le p$. Are there infinitely many cluster primes? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-17`, 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 17, maintained status record. Erdős Problems record 17, checked 2026-08-01. Problem 17; status field and linked bibliography https://www.erdosproblems.com/17
   - Also cited at Problem 17; 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 Infinitude of Cluster Primes: 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 17 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 Infinitude of Cluster Primes: 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 17. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/17.lean:L39; theorem erdos_17; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/17.lean#L39
   - 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 Infinitude of Cluster Primes: Supplies the pinned formal declaration whose human-readable editorial statement is published here.