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Erdős Problem #1063

OPENdatabase updated 2025-10-01

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 \(n_k\) be the least \(n \ge 2k\) such that all but one of the integers \(n - i\) with \(0 \le i < k\) divide \(\binom{n}{k}\).

Estimate \(n_k\) by finding a better upper bound than Cambie's \(n_k \leq k \cdot \operatorname{lcm}(1, \dotsc, k-1)\). The comparator takes its least common multiple in `ℕ` and casts the result. Writing the ascription as `((… ).lcm (fun n : ℕ => n) : ℝ)` instead puts it on the `Finset.lcm` application, so the coercion lands on `n` and the `lcm` is taken in `ℝ`, where `lcm` of non-zero elements is `1` and the whole comparator collapses to `k`.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 6 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1063.
Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA389360
Original pageerdosproblems.com/1063

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FULL WRITE-UP BY THIS SITE · JUL 25, 2026

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erdosproblems.com's own statement of #1063 is just 'estimate n_k' — open-ended, and it stays open here. What we actually proved: n_k = exp(o(k)), a genuine improvement on Cambie's exp((1+o(1))k) bound, which fills one specific formalized open target (a Google DeepMind Lean placeholder). A reader correctly called out an earlier version of this entry for reading like a full solve with no checkable proof behind it — full corrected writeup and proof now public.

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