The board / number theory / #1093
Erdős Problem #1093
OPENdatabase updated 2025-10-18
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).
If defined, the deficiency is the count of \(0 \le i < k\) such that \(n - i\) is \(k\)-smooth, that is, divisible only by primes \(\le k\). `Nat.smoothNumbers m` is the set of positive naturals all of whose prime factors are \(< m\), so "\(k\)-smooth" in the sense above is `Nat.smoothNumbers (k + 1)`.
Are there infinitely many binomial coefficients with deficiency 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/1093.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1093
Attempts
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