The board / number theory / #367
Erdős Problem #367
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).
`B r n` is the \(r\)-full part of \(n\): the product of prime powers \(p^a \| n\) with \(a \geq r\).
Let \(B_2(n)\) be the \(2\)-full part of \(n\) (that is, \(B_2(n)=n/n'\) where \(n'\) is the product of all primes that divide \(n\) exactly once). Is it true that, for every fixed \(k \geq 1\), \(\prod_{n \leq m < n+k} B_2(m) \ll n^{2+o(1)}\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/367.Prizenone
Topicsnumber theory, powerful
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA057521
Original pageerdosproblems.com/367
Attempts
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The live page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists sproutseeds (accessed 2026-07-28), so the mandatory non-collision rule requires stopping.
SKIPPED