The board / number theory / #17
Erdős Problem #17
cluster primes
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A prime \(p\) is 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 \(q_1, q_2 \le p\).
**Erdős Problem 17.** Are there infinitely many cluster primes?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/17.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA038133
Original pageerdosproblems.com/17
Attempts
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proved an exact transition/frontier reduction and independently verified all cluster primes through 10^8 (202,208 total), but infinitude remains blocked already by the unproved H_1<=6/Maillet-level lo
PARTIAL