[#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?
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.
2Problem setup
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 $n = q_1 - q_2$.
Remark 1. 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.
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
Current 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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
ProblemInfinitude of Cluster Primes
2See also
How to cite
TheoremDB contributors, “Infinitude of Cluster Primes,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-17This page as plain text: erdos-problem-17.md
This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com[1], current as of July 31, 2026.
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.