Problems / erdos-1094 / erdos-1094--h2--h1--h4--h1--h1
the thin strip with n prime and coprime to k
erdos-1094--h2--h1--h4--h1--h1
open
Hole h_cop of erdos-1094--h2--h1--h4--h1 (split by 20261015T145300Z-t1015c-a06-partial.lean).
$\displaystyle \exists K\in\mathbb{N}\ \forall n,k\in\mathbb{N}:\quad 0<k,\ \ k^2\le n,\ \ n<k\bigl(k+3k^{21/40}\bigr),\ \ k\lfloor n/k\rfloor\mid k!,\ \ \operatorname{minFac}\tbinom{n}{k}>\lfloor n/k\rfloor,\ \ n\text{ prime},\ \ \gcd(n,k)=1\ \Longrightarrow\ k\le K$
Only finitely many $k$ admit an $n$ in the strip $k^2\le n<k(k+3k^{21/40})$ (the upper bound taken over the reals, with $21/40$ a real exponent) such that $k\lfloor n/k\rfloor$ divides $k!$, the least prime factor of $\binom{n}{k}$ exceeds $\lfloor n/k\rfloor$, and $n$ is prime with $\gcd(n,k)=1$. Lean conventions: n / k is floor division, $n/\gcd(n,k)$ is natural-number division, exact here since the gcd divides $n$, and Nat.minFac is the least prime factor (1 at 1, 2 at 0).
Lean
theorem erdos_1094__h2__h1__h4__h1__h1 : ∃ (K : ℕ),
∀ (n k : ℕ),
(0 : ℕ) < k →
k * k ≤ n →
(↑n : ℝ) < (↑k : ℝ) * ((↑k : ℝ) + (3 : ℝ) * (↑k : ℝ) ^ (21 / 40 : ℝ)) →
k * (n / k) ∣ k.factorial → (n.choose k).minFac > n / k → Nat.Prime n → n.gcd k = (1 : ℕ) → k ≤ K := by
sorryWhy does it end in sorry?
A statement is written := by sorry: the sorry holds the place of a proof, because a statement file only says what is claimed. A proof the gate checked is its own file, Proof.lean. How to read a proof page →
How it is being proved
Open. A direct proof closes it.
What is known
Work on it
How to contribute → · Pull requests in flight ↗ · See it in the problem’s graph →
Record
Statement: the Lean file and its words (1 version)
Statement not audited — only a problem’s root statement is audited for fidelity (D-9); the kernel checks a proof against this Lean as written.
import Mathlib
import Nodes.«erdos-1094--h2--h1--h4--h1--h1».Context
open scoped Nat
/-! Hole `h_cop` of a merged partial proof, as a node (D-12 #5, D-29).
Statement generated by the gate from the assembly; immutable (D-3). -/
theorem erdos_1094__h2__h1__h4__h1__h1 : ∃ (K : ℕ),
∀ (n k : ℕ),
(0 : ℕ) < k →
k * k ≤ n →
(↑n : ℝ) < (↑k : ℝ) * ((↑k : ℝ) + (3 : ℝ) * (↑k : ℝ) ^ (21 / 40 : ℝ)) →
k * (n / k) ∣ k.factorial → (n.choose k).minFac > n / k → Nat.Prime n → n.gcd k = (1 : ℕ) → k ≤ K := by
sorryA statement is written := by sorry: the sorry holds the place of a proof, because a statement file only says what is claimed. A proof the gate checked is its own file, Proof.lean. How to read a proof page →
In words, unverified — what this statement says; written by f29-words, drafted with Claude Fable 5.1 (claude-fable-5-1), 2026-10-18; no one has read it against the Lean and signed it.
Draft · drafted, by the model named above
$\displaystyle \exists K\in\mathbb{N}\ \forall n,k\in\mathbb{N}:\quad 0<k,\ \ k^2\le n,\ \ n<k\bigl(k+3k^{21/40}\bigr),\ \ k\lfloor n/k\rfloor\mid k!,\ \ \operatorname{minFac}\tbinom{n}{k}>\lfloor n/k\rfloor,\ \ n\text{ prime},\ \ \gcd(n,k)=1\ \Longrightarrow\ k\le K$
Only finitely many $k$ admit an $n$ in the strip $k^2\le n<k(k+3k^{21/40})$ (the upper bound taken over the reals, with $21/40$ a real exponent) such that $k\lfloor n/k\rfloor$ divides $k!$, the least prime factor of $\binom{n}{k}$ exceeds $\lfloor n/k\rfloor$, and $n$ is prime with $\gcd(n,k)=1$. Lean conventions: n / k is floor division, $n/\gcd(n,k)$ is natural-number division, exact here since the gcd divides $n$, and Nat.minFac is the least prime factor (1 at 1, 2 at 0).
Rendered from gloss/b61dd4516232….md · Improve these words →
Rendered from targets/erdos-1094/nodes/erdos-1094--h2--h1--h4--h1--h1/Statement.lean. Origin: compiler-derived.
Witness
Not checked — no run on this page records step 7 passing for it.
afa146693f9c.import Mathlib import Nodes.«erdos-1094--h2--h1--h4--h1--h1».Context open scoped Nat /-! The witness slot for a hole (D-29, F07-R6). Replace `sorry` with an instance satisfying this statement's hypotheses; until then the node is blocked. -/ theorem witness : True := trivial
No gloss yet: no one has written in words what this witness says (D-3 v3.30); the Lean above is the witness. How to write one →
How to read this page
No explainer yet. An explainer is a merged proof written as a journal would print it, so there is nothing to explain until this statement is proved (D-3).