The board / number theory / #849
Erdős Problem #849
Singmaster's conjecture
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 every integer \(t\geq1\), there is some integer \(a\) such that \({n \choose k} = a\) with \(1\leq k \le \frac{n}{2}\) has exactly \(t\) solutions?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/849.Attempts
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Exact modulo the published fixed-pair theorem, every \(\binom n2\) with \(4\leq n\leq10^{32}\) has half-triangle multiplicity at most 4 (only \(n=78\) attains 4); a 4.319-billion-candidate exact sieve
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