The board / number theory / #459
Erdős Problem #459
SOLVED LEANdatabase updated 2026-03-11
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The function from the problem, in its equivalent form: `f u` is the smallest `v > u` all of whose prime factors divide `u`. (Equivalently, `f u` is the largest `v` such that no `m ∈ (u, v)` is composed entirely of primes dividing `u * v`.)
Let \(f(u)\) be the largest \(v\) such that no \(m\in (u,v)\) is composed entirely of primes dividing \(uv\). Estimate \(f(u)\). The estimate \(u + 2 \le f(u) \le u^2\) holds for every \(u \ge 2\). The upper bound is attained when \(u\) is prime, and the lower bound when \(u = 2^k - 2\) with \(k \ge 2\); Cambie further showed that \(f(n) = (1 + o(1))n\) for almost all \(n\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/459.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanyes
OEISA289280
Original pageerdosproblems.com/459
Attempts
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