The board / number theory / #731
Erdős Problem #731
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).
The least integer \(m \geq 1\) such that \(m \nmid \binom{2n}{n}\), denoted \(A(n)\) in [EGRS75]. The set is nonempty, since it contains \(\binom{2n}{n}+1\) (see `leastNonDivisor_spec`), so `sInf` is the true minimum. The condition `0 < m` excludes `m = 0`.
Find some reasonable function \(f(n)\) such that, for almost all integers \(n\), the least integer \(m\) such that \(m\nmid \binom{2n}{n}\) satisfies \[m\sim f(n).\] Here "almost all" means natural density \(1\), and \(m \sim f(n)\) means that for every \(\varepsilon>0\) the set of \(n\) with \(|m/f(n)-1|<\varepsilon\) has density \(1\).
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/731.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA006197
Original pageerdosproblems.com/731
Attempts
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The live page is OPEN but lists 1 claimed proof—a full proof claimed by Eric Li (using GPT-5.5 Pro), submitted 2026-07-17—and lists “Currently working on this problem: None”; the claimed-proof stop co
SKIPPED