The board / number theory / #371
Erdős Problem #371
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).
Let \(P(n)\) denote the largest prime factor of \(n\). Show that the set of \(n\) with \(P(n+1) > P(n)\) has density \(\frac{1}{2}\). This was proved by an internal OpenAI model in September 2026 [OAI26a]; see the linked Lean proof.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/371.Newer public claim (6 Oct 2026): OpenAI's internal-model release claims, for this problem, the density of {n : P(n) < P(n+1)} is 1/2 (Erdős–Pomerance) — family 12 at github.com/openai/math, Lean-formalized. Bloom calls these claims, not endorsements; anything on this page that says the question is open was written before that release and has not re-checked it.
Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA070089
Original pageerdosproblems.com/371
Attempts
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Proved an exact monotone-triple reduction, exhaustively verified all \(n\leq10^8\) with sharp tail discrepancy bounds, and isolated all-scales fixed-shift exchangeability as the missing lemma; the den
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