The board / number theory / #648
Erdős Problem #648
SOLVED LEANdatabase updated 2026-02-04
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).
`g n` is the largest `t` such that there exist integers `2 ≤ a 0 < a 1 < ⋯ < a (t - 1) < n` whose greatest prime factors are strictly decreasing.
Let \(g(n)\) denote the largest \(t\) such that there exist integers \(2\leq a_1<a_2<\cdots <a_t <n\) such that \[P(a_1)>P(a_2)>\cdots >P(a_t)\] where \(P(m)\) is the greatest prime factor of \(m\). Estimate \(g(n)\). Stijn Cambie has proved [Ca25b] \[g(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}.\] Cambie further asks whether there exists a constant \(c\) such that \[g(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}.\] Cambie's proof shows that such a \(c\) must satisfy \(2\leq c\leq 2\sqrt{2}\). The sequence \(a_1<a_2<\cdots<a_t\) is packaged as a strictly monotone map `a : Fin t → ℕ` with \(2\leq a_i<n\), the greatest prime factor \(P\) is `Nat.maxPrimeFac`, and \(g(n)\) is the supremum in `ℕ` of the achievable lengths \(t\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/648.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
OEISA391750
Original pageerdosproblems.com/648
Attempts
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