The board / number theory / #680
Erdős Problem #680
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).
Is it true that, for all sufficiently large \(n\), there exists some \(k\) such that \[ p(n+k)>k^2+1, \] where \(p(m)\) denotes the least prime factor of \(m\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/680.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/680
Attempts
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Exact reduction obtained; the first assertion holds for every \(116\le n\le10^9\) (next-prime witness), with exactly seven smaller exceptions, while the uniform square-root gap step and the exponentia
PARTIAL