[#P2972] Infinitude of Cluster Primes

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?

A finite mathematical diagram showing prime pairs and their even differences. — Prime pairs joined to their even differences.

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1Context

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.

2Definitions

Definition 1 (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 $n = q_1 - q_2$.

3What 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?

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 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]

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 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?	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 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] Relevance to this problem For erdos problem 17, pins the dated research frontier: 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. 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 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? Open full record →
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?	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 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] Relevance to this problem For Infinitude of Cluster Primes, record erdos-problem-17-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 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 $q_1, q_2 \le p$. 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 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? 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 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.

How the 2 records connect

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

ProblemInfinitude of Cluster Primes

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

“Infinitude of Cluster Primes.” TheoremDB. P2972. Problem statement; statement text SHA-256 dd107fadc9853ac2ad3bf2ea7e6ac804b5ccf237e3d5f9d55a4ca713135c0370. https://theoremdb.org/statement/?ref=P2972

BibTeX

@misc{theoremdb-problem-dd107fadc9853ac2ad3bf2ea7e6ac804b5ccf237e3d5f9d55a4ca713135c0370,
  title = {{Infinitude of Cluster Primes}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 dd107fadc9853ac2ad3bf2ea7e6ac804b5ccf237e3d5f9d55a4ca713135c0370},
  url = {https://theoremdb.org/statement/?ref=P2972}
}

Plain text: Built Markdown snapshot

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 17, maintained status record. Erdős Problems record 17, checked 2026-08-01. Problem 17; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 17 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 Infinitude of Cluster Primes: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 17. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/17.lean:L39; theorem erdos_17; 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 Infinitude of Cluster Primes: 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.

[#P2972] Infinitude of Cluster Primes

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