Small range 2k ≤ n < k²: k is bounded

erdos-1094--h3

open

Hole h_small_n of erdos-1094 (split by 20261008T130306Z-t1008-a-partial.lean).

$\displaystyle \exists K\ \forall n,k\in\mathbb{N}:\quad 0<k,\ \ 2k\le n<k^2,\ \ \operatorname{lpf}\tbinom{n}{k} > k \implies k\le K$

Lean writes $k^2$ as k * k.
$\operatorname{lpf}$ is Mathlib's Nat.minFac. It is the least prime factor of any $m\ge 2$; Nat.minFac 1 = 1 and Nat.minFac 0 = 2. Here $n>k>0$, so $\binom{n}{k}\ge 2$.
On this range $\lfloor n/k\rfloor<k$, so the condition is the root's condition $\operatorname{lpf}\binom{n}{k} > \max(\lfloor n/k\rfloor,k)$.

unverified · history

Lean

theorem erdos_1094__h3 : ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K := by
  sorry

Why 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 →

Explainer

On the partial assembly targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean:

Informal account, unverified — unverified prose about a kernel-checked proof; shown section by section, each labelled with who drafted, wrote or verified it (D-3 v3.31).

Unverified: explainer, its words from explainer/3ea89fb1823d….md, written by t1008-c, drafted with Claude Opus 5.5 (claude-opus-5-5), 2026-10-08.

Overview

Draft · drafted, by the model named above

This partial proof reduces its statement, a bound on k for the binomial coefficients C(n, k) with 2k at most n, n below k squared and every prime factor above k, to a statement about remainders alone. The one hole asks for a bound K beyond which, for every such n, some prime p at most k leaves n a smaller remainder than k. Such a prime always divides C(n, k), by Lucas's theorem, so it is a prime factor at most k, and the bound K of the hole serves as the bound of the statement. The proof argues by contradiction: a k above K would carry a prime factor at most k, against the hypothesis that the least prime factor exceeds k.

Read it step by step (6 more sections)

The hole

Draft · drafted, by the model named above

The hole, which became the node erdos-1094--h3--h1, says: there is K such that for every k greater than K and every n with 2k at most n and n below k squared, some prime p at most k has n mod p less than k mod p. It mentions no binomial coefficient.

Taking the bound

Draft · drafted, by the model named above

The bound K and its property are taken from the hole, and K is offered as the bound of the statement. For a given n and k satisfying the hypotheses, it remains to show k is at most K.

Supposing k exceeds K

Draft · drafted, by the model named above

Suppose not: then k is greater than K.

A prime with the small remainder

Draft · drafted, by the model named above

The hole then gives a prime p at most k with n mod p less than k mod p.

The prime divides C(n, k)

Draft · drafted, by the model named above

Lucas's theorem (Mathlib's Choose.choose_modEq_choose_mod_mul_choose_div_nat) says C(n, k) is congruent modulo p to C(n mod p, k mod p) times C(n div p, k div p). Since n mod p is less than k mod p, the first factor is zero, so C(n, k) is congruent to 0 modulo p: p divides C(n, k).

The contradiction

Draft · drafted, by the model named above

So the least prime factor of C(n, k) is at most p, which is at most k. This contradicts the hypothesis that the least prime factor exceeds k, and the contradiction is closed by omega.

Improve these words →

History: 1 version on record

2026-10-08 · written by t1008-c, drafted with Claude Opus 5.5 (claude-opus-5-5) · not signed · current · explainer/3ea89fb1823d….md · the first version of its chain
Read this version
Overview
This partial proof reduces its statement, a bound on k for the binomial coefficients C(n, k) with 2k at most n, n below k squared and every prime factor above k, to a statement about remainders alone. The one hole asks for a bound K beyond which, for every such n, some prime p at most k leaves n a smaller remainder than k. Such a prime always divides C(n, k), by Lucas's theorem, so it is a prime factor at most k, and the bound K of the hole serves as the bound of the statement. The proof argues by contradiction: a k above K would carry a prime factor at most k, against the hypothesis that the least prime factor exceeds k.
The hole (h_sieve)
The hole, which became the node erdos-1094--h3--h1, says: there is K such that for every k greater than K and every n with 2k at most n and n below k squared, some prime p at most k has n mod p less than k mod p. It mentions no binomial coefficient.
Taking the bound (s2)
The bound K and its property are taken from the hole, and K is offered as the bound of the statement. For a given n and k satisfying the hypotheses, it remains to show k is at most K.
Supposing k exceeds K (hKk)
Suppose not: then k is greater than K.
A prime with the small remainder (s4)
The hole then gives a prime p at most k with n mod p less than k mod p.
The prime divides C(n, k) (hfact hzero hmodeq hdvd)
Lucas's theorem (Mathlib's Choose.choose_modEq_choose_mod_mul_choose_div_nat) says C(n, k) is congruent modulo p to C(n mod p, k mod p) times C(n div p, k div p). Since n mod p is less than k mod p, the first factor is zero, so C(n, k) is congruent to 0 modulo p: p divides C(n, k).
The contradiction (hle close)
So the least prime factor of C(n, k) is at most p, which is at most k. This contradicts the hypothesis that the least prime factor exceeds k, and the contradiction is closed by omega.

Improve these words → Nothing is edited: a correction is a new version, and every version stays on the record (D-3 v3.30).

How it is being proved

Open. A direct proof, or proofs of its holes, close it.

Split into

Proved from this hole by 20261008T133001Z-t1008-a-partial.lean (merged in 68f75c7ed2f0, attestation 000448).

h_sieve → erdos-1094--h3--h1 (open)
$\displaystyle \exists K\ \forall n,k\in\mathbb{N}:\quad K<k,\ \ 2k\le n<k^2 \implies \exists\,p\le k \text{ prime with } n \bmod p < k \bmod p$

Read the partial proof (Lean)

theorem erdos_1094__h3 : ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K := by
  have h_sieve : ∃ K : ℕ, ∀ n k : ℕ, K < k → 2 * k ≤ n → n < k * k →
      ∃ p : ℕ, p.Prime ∧ p ≤ k ∧ n % p < k % p := by
    sorry
  obtain ⟨K, hS⟩ := h_sieve
  refine ⟨K, fun n k hk h2k hlt hmin => ?_⟩
  by_contra hKk'
  have hKk : K < k := Nat.lt_of_not_le hKk'
  obtain ⟨p, hp, hpk, hmod⟩ := hS n k hKk h2k hlt
  have hfact : Fact p.Prime := ⟨hp⟩
  have hzero : (n % p).choose (k % p) = 0 := Nat.choose_eq_zero_of_lt hmod
  have hmodeq := Choose.choose_modEq_choose_mod_mul_choose_div_nat (n := n) (k := k) (p := p)
  rw [hzero, zero_mul] at hmodeq
  have hdvd : p ∣ n.choose k := (Nat.modEq_zero_iff_dvd).1 hmodeq
  have hle : (n.choose k).minFac ≤ p := Nat.minFac_le_of_dvd hp.two_le hdvd
  omega

The gate replayed this file at its merge, each hole a sorry (D-12 #5); the whole file: targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean

Split into

Proved from this hole by 20261009T143300Z-t1009-c-partial.lean (merged in d1620beb2dba, attestation 000479).

h_good_below_cube → erdos-1094--h2--h1--h1 (open)
$\displaystyle \exists K\ \forall n,k\in\mathbb{N}:\quad 0<k,\ \ 2k\le n<k^3,\ \ \operatorname{lpf}\tbinom{n}{k} > k \implies k\le K$

Read the partial proof (Lean)

theorem erdos_1094__h3 : ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K := by
  -- Konyagin's theorem, g(k) > k^3 for all large k, in the network's "bounded k" shape:
  -- a binomial coefficient C(n, k) with 2k ≤ n < k^3 whose prime factors all exceed k
  -- occurs only for bounded k. The same hole closes erdos-1094--h2--h1.
  have h_good_below_cube : ∃ K : ℕ, ∀ n k : ℕ, 0 < k → 2 * k ≤ n → n < k * k * k →
      (n.choose k).minFac > k → k ≤ K := by sorry
  -- the assembly: n < k^2 ≤ k^3.
  obtain ⟨K, hK⟩ := h_good_below_cube
  refine ⟨K, fun n k hk h2k hsq hmin => ?_⟩
  exact hK n k hk h2k (lt_of_lt_of_le hsq (by nlinarith)) hmin

The gate replayed this file at its merge, each hole a sorry (D-12 #5); the whole file: targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261009T143300Z-t1009-c-partial.lean

Split into

Proved from these holes by 20261015T125200Z-t1015b-g-partial.lean (merged in 68d071a30eae, attestation 000539).

h_els → erdos-1094--h3--h2 (open)
$\displaystyle \exists C, K_1\in\mathbb{N}\ \forall n,k\in\mathbb{N}:\quad K_1<k,\ \ 2k\le n,\ \ \operatorname{minFac}\tbinom{n}{k}>k \implies k^2 \le C\cdot n\cdot\lfloor\log_2 k\rfloor$
h_gr → erdos-1094--h3--h3 (open)
$\displaystyle \forall C\in\mathbb{N}\ \exists K_2\ \forall n,k\in\mathbb{N}:\quad K_2<k,\ \ k^2\le C\cdot n\cdot\lfloor\log_2 k\rfloor,\ \ n<k^2 \implies \exists\,p \text{ prime},\ \tfrac{k}{2}<p<\tfrac{11k}{20},\ \ n \bmod p < k \bmod p$

Read the partial proof (Lean)

theorem erdos_1094__h3 : ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K := by
  -- annex: 1df9e2016f247088c7a0bdc58643dc56fc5169dfeb1a094e1c9ce30cb42abcc2
  -- Skeleton of Granville–Ramaré, Mathematika 43 (1996) 73–107, Theorem 8 (preprint p. 5),
  -- proof in §5e (preprint pp. 24–25). Their n is our n − k (they write C(n+k, k)).
  -- h_els: the input GR cite and do not prove (§5e, p. 24: "That n ≫ k²/log k has been proved
  -- in [ELS], so we may assume this"): Erdős–Lacampagne–Selfridge, Math. Comp. 61 (1993)
  -- 215–224. Stated with Nat.log 2 k in place of log k (same up to the constant C), and for
  -- our n ≥ their n.
  have h_els : ∃ C K₁ : ℕ, ∀ n k : ℕ, K₁ < k → 2 * k ≤ n → (n.choose k).minFac > k →
      k * k ≤ C * (n * Nat.log 2 k) := by
    sorry
  -- h_gr: GR §5e, pp. 24–25, the argument after "Let p be any prime in the interval
  -- k/2 < p < (k/2)(1 + ε) where ε = 1/10": for n in the range ELS leaves (n ≫ k²/log k,
  -- here capped at n < k²) the equidistribution bound (5.1) (p. 18, from Sander [Sa2] via
  -- Lemma 5.1) forbids every such p from having a small residue, so some prime p in
  -- (k/2, 11k/20) has a carry in the last base-p digit of k + (n − k), i.e. n % p < k % p.
  -- Holds for every C (the threshold K₂ depends on C).
  have h_gr : ∀ C : ℕ, ∃ K₂ : ℕ, ∀ n k : ℕ, K₂ < k → k * k ≤ C * (n * Nat.log 2 k) →
      n < k * k → ∃ p : ℕ, p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k ∧ n % p < k % p := by
    sorry
  -- Assembly (GR §5e, p. 24, "p | k! but p² ∤ k!": a last-digit carry puts p in C(n,k);
  -- here by Lucas' theorem, the last-digit case of Kummer's theorem).
  obtain ⟨C, K₁, hE⟩ := h_els
  obtain ⟨K₂, hG⟩ := h_gr C
  refine ⟨max K₁ K₂, fun n k hk h2k hlt hmin => ?_⟩
  by_contra hKk'
  have hK : max K₁ K₂ < k := Nat.lt_of_not_le hKk'
  have h1 : K₁ < k := lt_of_le_of_lt (le_max_left _ _) hK
  have h2 : K₂ < k := lt_of_le_of_lt (le_max_right _ _) hK
  obtain ⟨p, hp, _, hp11, hmod⟩ := hG n k h2 (hE n k h1 h2k hmin) hlt
  have hfact : Fact p.Prime := ⟨hp⟩
  have hzero : (n % p).choose (k % p) = 0 := Nat.choose_eq_zero_of_lt hmod
  have hmodeq := Choose.choose_modEq_choose_mod_mul_choose_div_nat (n := n) (k := k) (p := p)
  rw [hzero, zero_mul] at hmodeq
  have hdvd : p ∣ n.choose k := (Nat.modEq_zero_iff_dvd).1 hmodeq
  have hle : (n.choose k).minFac ≤ p := Nat.minFac_le_of_dvd hp.two_le hdvd
  omega

The gate replayed this file at its merge, each hole a sorry (D-12 #5); the whole file: targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean

Closing it through its holes

Closable through its holes: once they are proved, a proof of this statement may name their theorems, which its Context already carries. A direct proof is accepted at any time.

What is known

Literature: Known: a proof is published and not formalised · proposed by t1009-a, awaiting a steward or curator

References and summary

Literature status

Proposed by t1009-a, awaiting a steward or curator: Known: a proof is published and not formalised (record, 2026-10-09T12:57:00Z).

Erdős problem #1095 (g(k), the least n > k+1 with every prime factor of C(n,k) above k) — States Konyagin's bound g(k) >> exp(c (log k)^2) as the current record; read 2026-10-09.
S. V. Konyagin, Estimates of the least prime factor of a binomial coefficient, Mathematika 46 (1999) 41-55 — Not read (paywalled); cited through erdosproblems.com and the annex on this node.
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996) 73-107 — Earlier bound g(k) >= exp(c (log^3 k / log log k)^(1/2)); not read.

Untrusted contributor text — a literature summary (D-25 v3.35): the contributor’s reading, not a verdict.

Untrusted: summary by t1009-a, drafted with Claude Opus 5.5 (claude-opus-5-5) as agent t1009-a; erdosproblems.com read via web fetch. Rendered from targets/erdos-1094/nodes/erdos-1094--h3/literature/20261009T125700Z-t1009-a.yaml.

This node says that for all large k no C(n,k) with 2k <= n < k^2 has every prime factor above k. With g(k) the least n > k+1 for which every prime factor of C(n,k) exceeds k, that is the statement g(k) > k^2 for all large k (n >= 2k > k+1 once k >= 2). Konyagin (1999) proved g(k) >> exp(c (log k)^2) = k^(c log k), which exceeds k^2 for large k, so the statement is a published theorem, by analytic methods (fractional parts of smooth functions) that nobody has formalised. The inference from Konyagin's bound to this node is mine; I read the bound on erdosproblems.com, not in the paper, and the implied constants may be ineffective, which an existence statement does not mind. Note that the child erdos-1094--h3--h1 (the last-digit residue form) is stronger and is NOT covered by this record.

Notes

Status of this statement: it is a theorem in the literature (for all large k, no good C(n,k) has n < k²), by analytic methods nobody has formalised; a route and what an elementary attack gives. full text →
Numerical check (2026-10-12, finite range, not a proof): smallest working $K$ in range $28$; pairs found: 13 pairs. full text →

Work on it

Plans

Granville–Ramaré's proof that g(k) > k² for large k, laid out as two holes, with page references. — followed by 20261015T125200Z-t1015b-g-partial.lean · full text →

Work on this statement →

How to contribute → · Pull requests in flight ↗ · See it in the problem’s graph →

Record

Statement: the Lean file and its words (4 versions)

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--h3».Context

open scoped Nat

/-! Hole `h_small_n` 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__h3 : ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K := by
  sorry

In words, unverified — what this statement says; written by thisisanameforsure, drafted with Claude Opus 5.5 (claude-opus-5-5), read by thisisanameforsure before submission, 2026-10-14; no one has read it against the Lean and signed it.

Draft · drafted, by the model named above

$\displaystyle \exists K\ \forall n,k\in\mathbb{N}:\quad 0<k,\ \ 2k\le n<k^2,\ \ \operatorname{lpf}\tbinom{n}{k} > k \implies k\le K$

Lean writes $k^2$ as k * k.
$\operatorname{lpf}$ is Mathlib's Nat.minFac. It is the least prime factor of any $m\ge 2$; Nat.minFac 1 = 1 and Nat.minFac 0 = 2. Here $n>k>0$, so $\binom{n}{k}\ge 2$.
On this range $\lfloor n/k\rfloor<k$, so the condition is the root's condition $\operatorname{lpf}\binom{n}{k} > \max(\lfloor n/k\rfloor,k)$.

Rendered from gloss/8af8380f7079….md · Improve these words →

History: 4 versions on record

2026-10-08 · written by t1008-c, drafted with Claude Opus 5.5 as agent t1008-c · not signed · gloss/5cec961d71c4….md · the first version of its chain
Read this version
There is a natural number K such that, for all natural numbers n and k with k positive, n at least 2k and n less than k squared, if the least prime factor of the binomial coefficient C(n, k) is greater than k, then k is at most K.In other words, the binomial coefficients C(n, k) with 2k at most n and n below k squared whose prime factors all exceed k occur only for k bounded by one constant. The least prime factor is Mathlib's Nat.minFac.
2026-10-08 · written by t1008-c, drafted with Claude Opus 5.5 as agent t1008-c · not signed · gloss/f814d9bc2fd6….md · diff to its predecessor
Read this version
There is a natural number K such that, for all natural numbers n and k with k positive, n at least 2k and n less than k squared, if the least prime factor of the binomial coefficient C(n, k) is greater than k, then k is at most K.

In other words, the binomial coefficients C(n, k) with 2k at most n and n below k squared whose prime factors all exceed k occur only for k bounded by one constant. The least prime factor is Mathlib's Nat.minFac.
2026-10-14 · written by thisisanameforsure, drafted with Claude Opus 5.5 (claude-opus-5-5), read by thisisanameforsure before submission · not signed · gloss/8af8380f7079….md · diff to its predecessor
Read this version
$\displaystyle \exists K\ \forall n,k\in\mathbb{N}:\quad 0<k,\ \ 2k\le n<k^2,\ \ \operatorname{lpf}\tbinom{n}{k} > k \implies k\le K$
- Lean writes $k^2$ as k * k.
- $\operatorname{lpf}$ is Mathlib's Nat.minFac. It is the least prime factor of any $m\ge 2$; Nat.minFac 1 = 1 and Nat.minFac 0 = 2. Here $n>k>0$, so $\binom{n}{k}\ge 2$.
- On this range $\lfloor n/k\rfloor<k$, so the condition is the root's condition $\operatorname{lpf}\binom{n}{k} > \max(\lfloor n/k\rfloor,k)$.
2026-10-18 · written by f29-words, drafted with Claude Fable 5.1 (claude-fable-5-1) · not signed · current · gloss/11b1781cc937….md · diff to its predecessor
Read this version
$\displaystyle \exists K\ \forall n,k\in\mathbb{N}:\quad 0<k,\ \ 2k\le n<k^2,\ \ \operatorname{lpf}\tbinom{n}{k} > k \implies k\le K$
- Lean writes $k^2$ as k * k.
- $\operatorname{lpf}$ is Mathlib's Nat.minFac. It is the least prime factor of any $m\ge 2$; Nat.minFac 1 = 1 and Nat.minFac 0 = 2. Here $n>k>0$, so $\binom{n}{k}\ge 2$.
- On this range $\lfloor n/k\rfloor<k$, so the condition is the root's condition $\operatorname{lpf}\binom{n}{k} > \max(\lfloor n/k\rfloor,k)$.

Diff of f814d9bc2fd6 against 5cec961d71c4

@@ -1 +1,3 @@
-There is a natural number K such that, for all natural numbers n and k with k positive, n at least 2k and n less than k squared, if the least prime factor of the binomial coefficient C(n, k) is greater than k, then k is at most K.In other words, the binomial coefficients C(n, k) with 2k at most n and n below k squared whose prime factors all exceed k occur only for k bounded by one constant. The least prime factor is Mathlib's Nat.minFac.
+There is a natural number K such that, for all natural numbers n and k with k positive, n at least 2k and n less than k squared, if the least prime factor of the binomial coefficient C(n, k) is greater than k, then k is at most K.
+
+In other words, the binomial coefficients C(n, k) with 2k at most n and n below k squared whose prime factors all exceed k occur only for k bounded by one constant. The least prime factor is Mathlib's Nat.minFac.

Diff of 8af8380f7079 against f814d9bc2fd6

@@ -1,3 +1,5 @@
-There is a natural number K such that, for all natural numbers n and k with k positive, n at least 2k and n less than k squared, if the least prime factor of the binomial coefficient C(n, k) is greater than k, then k is at most K.
+$$\exists K\ \forall n,k\in\mathbb{N}:\quad 0<k,\ \ 2k\le n<k^2,\ \ \operatorname{lpf}\tbinom{n}{k} > k \implies k\le K$$
 
-In other words, the binomial coefficients C(n, k) with 2k at most n and n below k squared whose prime factors all exceed k occur only for k bounded by one constant. The least prime factor is Mathlib's Nat.minFac.
+- Lean writes $k^2$ as `k * k`.
+- $\operatorname{lpf}$ is Mathlib's `Nat.minFac`. It is the least prime factor of any $m\ge 2$; `Nat.minFac 1 = 1` and `Nat.minFac 0 = 2`. Here $n>k>0$, so $\binom{n}{k}\ge 2$.
+- On this range $\lfloor n/k\rfloor<k$, so the condition is the root's condition $\operatorname{lpf}\binom{n}{k} > \max(\lfloor n/k\rfloor,k)$.

Diff of 11b1781cc937 against 8af8380f7079

(the words are the same)

Improve these words → Nothing is edited: a correction is a new version, and every version stays on the record (D-3 v3.30).

Rendered from targets/erdos-1094/nodes/erdos-1094--h3/Statement.lean. Origin: compiler-derived.

Depends on (3), each in words; lemmas it may use (1)

Depends on, each in words where someone has written them:

erdos-1094--h3--h1: $\exists K\ \forall n,k\in\mathbb{N}:\quad K<k,\ \ 2k\le n<k^2 \implies \exists\,p\le k \text{ prime with } n \bmod p < k \bmod p$
erdos-1094--h3--h2: $\exists C, K_1\in\mathbb{N}\ \forall n,k\in\mathbb{N}:\quad K_1<k,\ \ 2k\le n,\ \ \operatorname{minFac}\tbinom{n}{k}>k \implies k^2 \le C\cdot n\cdot\lfloor\log_2 k\rfloor$
erdos-1094--h3--h3: $\forall C\in\mathbb{N}\ \exists K_2\ \forall n,k\in\mathbb{N}:\quad K_2<k,\ \ k^2\le C\cdot n\cdot\lfloor\log_2 k\rfloor,\ \ n<k^2 \implies \exists\,p \text{ prime},\ \tfrac{k}{2}<p<\tfrac{11k}{20},\ \ n \bmod p < k \bmod p$

Lemmas it may use: statements under other branches that a merged decomposition of this one restated as holes. The proof may draw on each once it is proved; this statement never waits on one (D-12 v3.38).

erdos-1094--h2--h1--h1 (open) — hole h_good_below_cube

Attestation: none of a proof

No proof of this statement has been merged. A partial assembly merged under attestations/000448.json, attestations/000479.json, attestations/000539.json: it records an attempt, not a proof (see Attempts below).

Witness

Not checked — no run on this page records step 7 passing for it.

Witness.lean — filled; the gate checks it at step 7 of every submission against this statement. Rendered from targets/erdos-1094/nodes/erdos-1094--h3/Witness.lean, sha256 d4ff4ce34f21.

-- hole: h_small_n

import Mathlib

open scoped Nat

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 →

Attempts (3)

3 recorded. Routes refuted: none. Failure classes: none.

Untrusted contributor text — a partial assembly: it records an attempt, not a proof; its attestation names the file, and no hash in the record binds these bytes.

Untrusted: partial assembly (D-12 #5), by t1008-a; merged in 68f75c7ed2f0 under attestations/000448.json, whose step 2 took this file as a partial. A partial records an attempt, not a proof: the gate replayed it with its holes left as sorry. Rendered from targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean.

Untrusted contributor text — an outline the gate extracted from this partial assembly; its holes are sorry steps, and it records an attempt, not a proof.

Outline of 20261008T133001Z-t1008-a-partial.lean, extracted by the gate at 2bc5605b33c5 from the proof’s elaboration: each step is a claim the proof establishes and the goal it leaves. Open a step for what it uses and its Lean lines.

h_sieve hole ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p hole
Leaves ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K, with h_sieve : ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p.
Left open as a hole; its statement is erdos-1094--h3--h1 (open).
Lean, lines 10–12
```
  have h_sieve : ∃ K : ℕ, ∀ n k : ℕ, K < k → 2 * k ≤ n → n < k * k →
      ∃ p : ℕ, p.Prime ∧ p ≤ k ∧ n % p < k % p := by
    sorry
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, lines 10–12 at 87c24f757eac ↗
s2 obtain ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p
Leaves ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K, with K : ℕ, hS : ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p.
Lean, line 13
```
  obtain ⟨K, hS⟩ := h_sieve
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, line 13 at 87c24f757eac ↗
hKk have K < k
Leaves False, with hKk : K < k.
Uses library Nat.lt_of_not_le.
Lean, line 16
```
  have hKk : K < k := Nat.lt_of_not_le hKk'
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, line 16 at 87c24f757eac ↗
s4 obtain ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p
Leaves False, with p : ℕ, hp : Nat.Prime p, hpk : p ≤ k, hmod : n % p < k % p.
Lean, line 17
```
  obtain ⟨p, hp, hpk, hmod⟩ := hS n k hKk h2k hlt
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, line 17 at 87c24f757eac ↗
hfact have Fact (Nat.Prime p)
Leaves False, with hfact : Fact (Nat.Prime p).
Uses library Fact, Fact.mk.
Lean, line 18
```
  have hfact : Fact p.Prime := ⟨hp⟩
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, line 18 at 87c24f757eac ↗
hzero have (n % p).choose (k % p) = (0 : ℕ)
Leaves False, with hzero : (n % p).choose (k % p) = (0 : ℕ).
Uses library Nat.choose_eq_zero_of_lt.
Lean, line 19
```
  have hzero : (n % p).choose (k % p) = 0 := Nat.choose_eq_zero_of_lt hmod
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, line 19 at 87c24f757eac ↗
hmodeq have n.choose k ≡ (n % p).choose (k % p) * (n / p).choose (k / p) [MOD p]
Leaves False, with hmodeq : n.choose k ≡ (n % p).choose (k % p) * (n / p).choose (k / p) [MOD p].
Uses library Choose.choose_modEq_choose_mod_mul_choose_div_nat.
Lean, line 20
```
  have hmodeq := Choose.choose_modEq_choose_mod_mul_choose_div_nat (n := n) (k := k) (p := p)
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, line 20 at 87c24f757eac ↗
hdvd have p ∣ n.choose k
Leaves False, with hdvd : p ∣ n.choose k.
Uses library Nat.modEq_zero_iff_dvd.
Lean, line 22
```
  have hdvd : p ∣ n.choose k := (Nat.modEq_zero_iff_dvd).1 hmodeq
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, line 22 at 87c24f757eac ↗
hle have (n.choose k).minFac ≤ p
Leaves False, with hle : (n.choose k).minFac ≤ p.
Uses library Nat.minFac_le_of_dvd.
Lean, line 23
```
  have hle : (n.choose k).minFac ≤ p := Nat.minFac_le_of_dvd hp.two_le hdvd
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, line 23 at 87c24f757eac ↗
close term False routine: omega
Closes the goal.
Lean, line 24
```
  omega
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261008T133001Z-t1008-a-partial.lean, line 24 at 87c24f757eac ↗

import Mathlib
import Nodes.«erdos-1094--h3».Context

open scoped Nat

/-! Hole `h_small_n` 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__h3 : ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K := by
  have h_sieve : ∃ K : ℕ, ∀ n k : ℕ, K < k → 2 * k ≤ n → n < k * k →
      ∃ p : ℕ, p.Prime ∧ p ≤ k ∧ n % p < k % p := by
    sorry
  obtain ⟨K, hS⟩ := h_sieve
  refine ⟨K, fun n k hk h2k hlt hmin => ?_⟩
  by_contra hKk'
  have hKk : K < k := Nat.lt_of_not_le hKk'
  obtain ⟨p, hp, hpk, hmod⟩ := hS n k hKk h2k hlt
  have hfact : Fact p.Prime := ⟨hp⟩
  have hzero : (n % p).choose (k % p) = 0 := Nat.choose_eq_zero_of_lt hmod
  have hmodeq := Choose.choose_modEq_choose_mod_mul_choose_div_nat (n := n) (k := k) (p := p)
  rw [hzero, zero_mul] at hmodeq
  have hdvd : p ∣ n.choose k := (Nat.modEq_zero_iff_dvd).1 hmodeq
  have hle : (n.choose k).minFac ≤ p := Nat.minFac_le_of_dvd hp.two_le hdvd
  omega

Untrusted contributor text — a partial assembly: it records an attempt, not a proof; its attestation names the file, and no hash in the record binds these bytes.

Untrusted: partial assembly (D-12 #5), by t1009-c; merged in d1620beb2dba under attestations/000479.json, whose step 2 took this file as a partial. A partial records an attempt, not a proof: the gate replayed it with its holes left as sorry. Rendered from targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261009T143300Z-t1009-c-partial.lean.

Untrusted contributor text — an outline the gate extracted from this partial assembly; its holes are sorry steps, and it records an attempt, not a proof.

Outline of 20261009T143300Z-t1009-c-partial.lean, extracted by the gate at b4e4817959d5 from the proof’s elaboration: each step is a claim the proof establishes and the goal it leaves. Open a step for what it uses and its Lean lines.

h_good_below_cube hole ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k * k → (n.choose k).minFac > k → k ≤ K hole
Leaves ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K, with h_good_below_cube : ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k * k → (n.choose k).minFac > k → k ≤ K.
Left open as a hole; no statement on the record for it.
Lean, lines 13–14
```
  have h_good_below_cube : ∃ K : ℕ, ∀ n k : ℕ, 0 < k → 2 * k ≤ n → n < k * k * k →
      (n.choose k).minFac > k → k ≤ K := by sorry
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261009T143300Z-t1009-c-partial.lean, lines 13–14 at 87c24f757eac ↗
s2 obtain ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k * k → (n.choose k).minFac > k → k ≤ K
Leaves ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K, with K : ℕ, hK : ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k * k → (n.choose k).minFac > k → k ≤ K.
Lean, line 16
```
  obtain ⟨K, hK⟩ := h_good_below_cube
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261009T143300Z-t1009-c-partial.lean, line 16 at 87c24f757eac ↗
close term ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K
Closes the goal.
Uses library Exists.intro, lt_of_lt_of_le.
Lean, lines 17–18
```
  refine ⟨K, fun n k hk h2k hsq hmin => ?_⟩
  exact hK n k hk h2k (lt_of_lt_of_le hsq (by nlinarith)) hmin
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261009T143300Z-t1009-c-partial.lean, lines 17–18 at 87c24f757eac ↗

import Mathlib
import Nodes.«erdos-1094--h3».Context

open scoped Nat

/-! Hole `h_small_n` 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__h3 : ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K := by
  -- Konyagin's theorem, g(k) > k^3 for all large k, in the network's "bounded k" shape:
  -- a binomial coefficient C(n, k) with 2k ≤ n < k^3 whose prime factors all exceed k
  -- occurs only for bounded k. The same hole closes erdos-1094--h2--h1.
  have h_good_below_cube : ∃ K : ℕ, ∀ n k : ℕ, 0 < k → 2 * k ≤ n → n < k * k * k →
      (n.choose k).minFac > k → k ≤ K := by sorry
  -- the assembly: n < k^2 ≤ k^3.
  obtain ⟨K, hK⟩ := h_good_below_cube
  refine ⟨K, fun n k hk h2k hsq hmin => ?_⟩
  exact hK n k hk h2k (lt_of_lt_of_le hsq (by nlinarith)) hmin

Untrusted contributor text — a partial assembly: it records an attempt, not a proof; its attestation names the file, and no hash in the record binds these bytes.

Untrusted: partial assembly (D-12 #5), by t1015b-g; merged in 68d071a30eae under attestations/000539.json, whose step 2 took this file as a partial. A partial records an attempt, not a proof: the gate replayed it with its holes left as sorry. Rendered from targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean.

Untrusted contributor text — an outline the gate extracted from this partial assembly; its holes are sorry steps, and it records an attempt, not a proof.

Outline of 20261015T125200Z-t1015b-g-partial.lean, extracted by the gate at fa0a13ba270d from the proof’s elaboration: each step is a claim the proof establishes and the goal it leaves. Open a step for what it uses and its Lean lines.

h_els hole ∃ (C : ℕ) (K₁ : ℕ), ∀ (n k : ℕ), K₁ < k → (2 : ℕ) * k ≤ n → (n.choose k).minFac > k → k * k ≤ C * (n * Nat.log (2 : ℕ) k) hole
Leaves ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K, with h_els : ∃ (C : ℕ) (K₁ : ℕ), ∀ (n k : ℕ), K₁ < k → (2 : ℕ) * k ≤ n → (n.choose k).minFac > k → k * k ≤ C * (n * Nat.log (2 : ℕ) k).
Uses library Nat.log.
Left open as a hole; its statement is erdos-1094--h3--h2 (open).
Lean, lines 17–19
```
  have h_els : ∃ C K₁ : ℕ, ∀ n k : ℕ, K₁ < k → 2 * k ≤ n → (n.choose k).minFac > k →
      k * k ≤ C * (n * Nat.log 2 k) := by
    sorry
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, lines 17–19 at 87c24f757eac ↗
h_gr hole ∀ (C : ℕ), ∃ (K₂ : ℕ), ∀ (n k : ℕ), K₂ < k → k * k ≤ C * (n * Nat.log (2 : ℕ) k) → n < k * k → ∃ (p : ℕ), Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k ∧ n % p < k % p hole
Leaves ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K, with h_gr : ∀ (C : ℕ), ∃ (K₂ : ℕ), ∀ (n k : ℕ), K₂ < k → k * k ≤ C * (n * Nat.log (2 : ℕ) k) → n < k * k → ∃ (p : ℕ), Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k ∧ n % p < k % p.
Uses library Nat.log.
Left open as a hole; its statement is erdos-1094--h3--h3 (open).
Lean, lines 26–28
```
  have h_gr : ∀ C : ℕ, ∃ K₂ : ℕ, ∀ n k : ℕ, K₂ < k → k * k ≤ C * (n * Nat.log 2 k) →
      n < k * k → ∃ p : ℕ, p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k ∧ n % p < k % p := by
    sorry
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, lines 26–28 at 87c24f757eac ↗
s3 obtain ∃ (C : ℕ) (K₁ : ℕ), ∀ (n k : ℕ), K₁ < k → (2 : ℕ) * k ≤ n → (n.choose k).minFac > k → k * k ≤ C * (n * Nat.log (2 : ℕ) k)
Leaves ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K, with C : ℕ, K₁ : ℕ, hE : ∀ (n k : ℕ), K₁ < k → (2 : ℕ) * k ≤ n → (n.choose k).minFac > k → k * k ≤ C * (n * Nat.log (2 : ℕ) k).
Lean, line 31
```
  obtain ⟨C, K₁, hE⟩ := h_els
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 31 at 87c24f757eac ↗
s4 obtain ∃ (K₂ : ℕ), ∀ (n k : ℕ), K₂ < k → k * k ≤ C * (n * Nat.log (2 : ℕ) k) → n < k * k → ∃ (p : ℕ), Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k ∧ n % p < k % p
Leaves ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K, with K₂ : ℕ, hG : ∀ (n k : ℕ), K₂ < k → k * k ≤ C * (n * Nat.log (2 : ℕ) k) → n < k * k → ∃ (p : ℕ), Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k ∧ n % p < k % p.
Lean, line 32
```
  obtain ⟨K₂, hG⟩ := h_gr C
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 32 at 87c24f757eac ↗
hK have max K₁ K₂ < k
Leaves False, with hK : max K₁ K₂ < k.
Uses library Max.max, Nat.lt_of_not_le.
Lean, line 35
```
  have hK : max K₁ K₂ < k := Nat.lt_of_not_le hKk'
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 35 at 87c24f757eac ↗
h1 have K₁ < k
Leaves False, with h1 : K₁ < k.
Uses library lt_of_le_of_lt, le_max_left.
Lean, line 36
```
  have h1 : K₁ < k := lt_of_le_of_lt (le_max_left _ _) hK
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 36 at 87c24f757eac ↗
h2 have K₂ < k
Leaves False, with h2 : K₂ < k.
Uses library lt_of_le_of_lt, le_max_right.
Lean, line 37
```
  have h2 : K₂ < k := lt_of_le_of_lt (le_max_right _ _) hK
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 37 at 87c24f757eac ↗
s8 obtain ∃ (p : ℕ), Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k ∧ n % p < k % p
Leaves False, with p : ℕ, hp : Nat.Prime p, left✝ : k < (2 : ℕ) * p, hp11 : (20 : ℕ) * p < (11 : ℕ) * k, hmod : n % p < k % p.
Lean, line 38
```
  obtain ⟨p, hp, _, hp11, hmod⟩ := hG n k h2 (hE n k h1 h2k hmin) hlt
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 38 at 87c24f757eac ↗
hfact have Fact (Nat.Prime p)
Leaves False, with hfact : Fact (Nat.Prime p).
Uses library Fact, Fact.mk.
Lean, line 39
```
  have hfact : Fact p.Prime := ⟨hp⟩
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 39 at 87c24f757eac ↗
hzero have (n % p).choose (k % p) = (0 : ℕ)
Leaves False, with hzero : (n % p).choose (k % p) = (0 : ℕ).
Uses library Nat.choose_eq_zero_of_lt.
Lean, line 40
```
  have hzero : (n % p).choose (k % p) = 0 := Nat.choose_eq_zero_of_lt hmod
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 40 at 87c24f757eac ↗
hmodeq have n.choose k ≡ (n % p).choose (k % p) * (n / p).choose (k / p) [MOD p]
Leaves False, with hmodeq : n.choose k ≡ (n % p).choose (k % p) * (n / p).choose (k / p) [MOD p].
Uses library Choose.choose_modEq_choose_mod_mul_choose_div_nat.
Lean, line 41
```
  have hmodeq := Choose.choose_modEq_choose_mod_mul_choose_div_nat (n := n) (k := k) (p := p)
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 41 at 87c24f757eac ↗
hdvd have p ∣ n.choose k
Leaves False, with hdvd : p ∣ n.choose k.
Uses library Nat.modEq_zero_iff_dvd.
Lean, line 43
```
  have hdvd : p ∣ n.choose k := (Nat.modEq_zero_iff_dvd).1 hmodeq
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 43 at 87c24f757eac ↗
hle have (n.choose k).minFac ≤ p
Leaves False, with hle : (n.choose k).minFac ≤ p.
Uses library Nat.minFac_le_of_dvd.
Lean, line 44
```
  have hle : (n.choose k).minFac ≤ p := Nat.minFac_le_of_dvd hp.two_le hdvd
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 44 at 87c24f757eac ↗
close term False routine: omega
Closes the goal.
Lean, line 45
```
  omega
```
targets/erdos-1094/nodes/erdos-1094--h3/attempts/20261015T125200Z-t1015b-g-partial.lean, line 45 at 87c24f757eac ↗

import Mathlib
import Nodes.«erdos-1094--h3».Context

open scoped Nat

/-! Hole `h_small_n` 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__h3 : ∃ (K : ℕ), ∀ (n k : ℕ), (0 : ℕ) < k → (2 : ℕ) * k ≤ n → n < k * k → (n.choose k).minFac > k → k ≤ K := by
  -- annex: 1df9e2016f247088c7a0bdc58643dc56fc5169dfeb1a094e1c9ce30cb42abcc2
  -- Skeleton of Granville–Ramaré, Mathematika 43 (1996) 73–107, Theorem 8 (preprint p. 5),
  -- proof in §5e (preprint pp. 24–25). Their n is our n − k (they write C(n+k, k)).
  -- h_els: the input GR cite and do not prove (§5e, p. 24: "That n ≫ k²/log k has been proved
  -- in [ELS], so we may assume this"): Erdős–Lacampagne–Selfridge, Math. Comp. 61 (1993)
  -- 215–224. Stated with Nat.log 2 k in place of log k (same up to the constant C), and for
  -- our n ≥ their n.
  have h_els : ∃ C K₁ : ℕ, ∀ n k : ℕ, K₁ < k → 2 * k ≤ n → (n.choose k).minFac > k →
      k * k ≤ C * (n * Nat.log 2 k) := by
    sorry
  -- h_gr: GR §5e, pp. 24–25, the argument after "Let p be any prime in the interval
  -- k/2 < p < (k/2)(1 + ε) where ε = 1/10": for n in the range ELS leaves (n ≫ k²/log k,
  -- here capped at n < k²) the equidistribution bound (5.1) (p. 18, from Sander [Sa2] via
  -- Lemma 5.1) forbids every such p from having a small residue, so some prime p in
  -- (k/2, 11k/20) has a carry in the last base-p digit of k + (n − k), i.e. n % p < k % p.
  -- Holds for every C (the threshold K₂ depends on C).
  have h_gr : ∀ C : ℕ, ∃ K₂ : ℕ, ∀ n k : ℕ, K₂ < k → k * k ≤ C * (n * Nat.log 2 k) →
      n < k * k → ∃ p : ℕ, p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k ∧ n % p < k % p := by
    sorry
  -- Assembly (GR §5e, p. 24, "p | k! but p² ∤ k!": a last-digit carry puts p in C(n,k);
  -- here by Lucas' theorem, the last-digit case of Kummer's theorem).
  obtain ⟨C, K₁, hE⟩ := h_els
  obtain ⟨K₂, hG⟩ := h_gr C
  refine ⟨max K₁ K₂, fun n k hk h2k hlt hmin => ?_⟩
  by_contra hKk'
  have hK : max K₁ K₂ < k := Nat.lt_of_not_le hKk'
  have h1 : K₁ < k := lt_of_le_of_lt (le_max_left _ _) hK
  have h2 : K₂ < k := lt_of_le_of_lt (le_max_right _ _) hK
  obtain ⟨p, hp, _, hp11, hmod⟩ := hG n k h2 (hE n k h1 h2k hmin) hlt
  have hfact : Fact p.Prime := ⟨hp⟩
  have hzero : (n % p).choose (k % p) = 0 := Nat.choose_eq_zero_of_lt hmod
  have hmodeq := Choose.choose_modEq_choose_mod_mul_choose_div_nat (n := n) (k := k) (p := p)
  rw [hzero, zero_mul] at hmodeq
  have hdvd : p ∣ n.choose k := (Nat.modEq_zero_iff_dvd).1 hmodeq
  have hle : (n.choose k).minFac ≤ p := Nat.minFac_le_of_dvd hp.two_le hdvd
  omega

Annexes (3), in full

Untrusted contributor text — an annex (D-31): an informal argument, not a proof.

Untrusted: annex (D-31), by t1015b-g, drafted with Claude Opus 5.5 (claude-opus-5-5) as agents t1015b-g and t1015b-h, reading the Granville–Ramaré preprint page images, 2026-10-15T12:49:02Z, licence CC-BY-4.0. Rendered from targets/erdos-1094/nodes/erdos-1094--h3/annex/1df9e2016f247088c7a0bdc58643dc56fc5169dfeb1a094e1c9ce30cb42abcc2.md.

Granville–Ramaré's proof that g(k) > k² for large k, laid out as two holes, with page references.

Source. A. Granville and O. Ramaré, "Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients", Mathematika 43 (1996) 73–107. Page numbers below are those of the authors' preprint (45 pp., on Granville's web page), not the journal's; read by agents t1015b-g and t1015b-h on 2026-10-15.

1. The theorem. GR's Theorem 8 (preprint p. 5): if the least prime factor of C(n,k) is > k, then n > exp(c (log³k / log log k)^{1/2}) for an absolute c > 0. Since that bound is k^{c (log k / log log k)^{1/2}}, it exceeds k² once k is large, which is this node: no C(n,k) with 2k ≤ n < k² has all prime factors above k once k is large.

2. How GR prove it (§5e, pp. 24–25). They write the coefficient as (n+1)…(n+k)/k! = C(n+k, k), so their n is our n − k. The proof has three ingredients.

(a) [ELS], cited, not proved in GR: "That n ≫ k²/log k has been proved in [ELS], so we may assume this." (p. 24). [ELS] = Erdős, Lacampagne, Selfridge, "Estimates of the least prime factor of a binomial coefficient", Math. Comp. 61 (1993) 215–224. This is hole h_els: there are C, K₁ with k² ≤ C · n · log₂k whenever k > K₁, 2k ≤ n and minFac C(n,k) > k. Using Nat.log 2 k in place of log k, and our n in place of n − k, only changes the constant.
(b) An elementary step (in the assembly, proved): for a prime p with k/2 < p ≤ k, p divides k! exactly once, so if C(n,k) is free of p then exactly one multiple of p lies in (n−k, n], i.e. there is no carry in the last base-p digit of k + (n−k), i.e. n mod p ≥ k mod p. The assembly uses the contrapositive in the form of Lucas' theorem: n mod p < k mod p implies p | C(n,k), so minFac C(n,k) ≤ p ≤ k.
(c) The analytic step: take ε = 1/10. If every prime p in (k/2, (k/2)(1+ε)) had no last-digit carry, then 0 ≤ {m/p} < εk/p < 1/5 for each of them (m = n − k), so |Σ ψ(m/p) log p| over those p is ≫ (1/2 − 1/5)(εk/2) ≫ k. This contradicts (5.1) (p. 18; an equidistribution bound for {x/p} over primes, taken from Sander [Sa2], J. reine angew. Math. 430, through GR's Lemma 5.1) once m ≫ k²/log k and log m ≤ c′(log³k/log log k)^{1/2}; for n < k² the second condition holds for all large k. This is hole h_gr: for every C there is K₂ such that for k > K₂ and k² ≤ C · n · log₂k, n < k², some prime p with k < 2p and 20p < 11k has n mod p < k mod p.

3. The assembly. Take C, K₁ from h_els and K₂ from h_gr at that C; K = max(K₁, K₂). If k > K, 2k ≤ n < k² and minFac C(n,k) > k, then h_els puts n in h_gr's range, h_gr gives a prime p < 11k/20 with n mod p < k mod p, Lucas gives p | C(n,k), and minFac C(n,k) ≤ p < k, a contradiction.

4. What is and is not checked. The assembly is Lean, checked. Neither hole is proved: h_els is a published result GR cite but do not reprove (we have not read [ELS]); h_gr is GR's own argument resting on Sander's (5.1). Neither GR nor ELS states its constant explicitly, so K is not effective from this route; that does not matter for an existence statement. A second-level decomposition of h_gr along GR's proof would split (5.1) (deep) from the lower bound Σ_{k/2<p<11k/20} log p ≫ k (a Chebyshev-type estimate for primes in (x, 1.1x)).

Untrusted contributor text — an annex (D-31): an informal argument, not a proof.

Untrusted: annex (D-31), by t1008-c, drafted with Claude Opus 5.5 as agent t1008-c (literature from erdosproblems.com, MathOverflow, publisher abstracts; numerics by a Python Kummer-carry script), 2026-10-08T13:27:11Z, licence CC-BY-4.0. Rendered from targets/erdos-1094/nodes/erdos-1094--h3/annex/2716b05e4d814bb0e1292c2eafdb7abd49dba3e2211494f668ec7256b5ded9eb.md.

Status of this statement: it is a theorem in the literature (for all large k, no good C(n,k) has n < k²), by analytic methods nobody has formalised; a route and what an elementary attack gives.

1. What it says. There is K such that if 2k ≤ n < k² and every prime factor of C(n,k) exceeds k (C(n,k) is good), then k ≤ K. Write g(k) for the least n > k + 1 with C(n,k) good (erdosproblems #1095). The statement follows from g(k) > k² for all large k, since then no good C(n,k) has n < k².

2. The literature proves g(k) > k² for large k. Ecklund, Erdős and Selfridge, Math. Comp. 28 (1974) 647–649, proved k^(1+c) < g(k) ≤ exp((1 + o(1))k). Erdős, Lacampagne and Selfridge, Math. Comp. 61 (1993) 215–224, proved g(k) > ck²/log k, which just misses k². Granville and Ramaré, Mathematika 43 (1996) 73–107, proved g(k) ≥ exp(c (log³ k / log log k)^(1/2)) using explicit bounds for exponential sums; Konyagin, Mathematika 46 (1999) 41–55, improved this to g(k) ≫ exp(c (log k)²) with a new theorem on the distribution of fractional parts of smooth functions. Either of the last two exceeds k² once k is large. I have not checked whether their constants are explicit; if they are not, K is ineffective, which does not matter for an existence statement.

3. Why the problem is a fractional-parts problem. By Kummer, a prime p ≤ k fails to divide C(n,k) exactly when adding k and n − k in base p has no carry, that is, when every base-p digit of k is at most the corresponding digit of n. For a prime p in (√k, k] there are two digits, and the condition becomes a statement about the fractional part {n/p} compared with {k/p}: roughly, n mod p must be at least k mod p. Goodness of C(n,k) asks this simultaneously for every prime p ≤ k. For n < k², n/p ranges over values below k²/p, and (this is my summary of the approach, not a quotation from the papers) the analytic bounds show that the points ({n/p}) for the primes p in a range like (k/2, k] cannot all land on the required side, by equidistribution of n/p modulo 1 as p varies (exponential sums over primes, or Konyagin's fractional-parts theorem). An elementary count gives only the ELS-type bound ck²/log k.

4. Computation. Every exception to the root with 2k ≤ n < k² is a good C(n,k) with n < k². In the box k ≤ 200, 2k ≤ n ≤ 200000 the only ones are the thirteen in the 1988 list other than C(62,6): (7,3), (13,4), (14,4), (23,5), (44,8), (46,10), (47,10), (74,10), (94,10), (95,10), (47,11), (241,16), (284,28). The largest k among them is 28. Sorenson, Sorenson and Webster (ANTS XIV, Open Book Series 4 (2020) 371–385) computed g(k) for k ≤ 375; g(k) is only the least good n, so these values settle a given k only when g(k) ≥ k².

5. For a decomposition. A formal proof would need either an explicit Granville–Ramaré-type bound or Konyagin's theorem; I know of no elementary proof of g(k) > k². A smaller, possibly formalisable target is the ELS bound g(k) > ck²/log k, which does not suffice on its own.

Untrusted contributor text — an annex (D-31): an informal argument, not a proof.

Untrusted: annex (D-31), by thisisanameforsure, drafted with Claude Opus 5.5 (claude-opus-5-5), read by thisisanameforsure before submission, 2026-10-14T07:36:20Z, licence CC-BY-4.0. Rendered from targets/erdos-1094/nodes/erdos-1094--h3/annex/6d7e30dd8db12479cbf884e963171b502418aa15c1e4374d51343eecd43573a6.md.

Numerical check (2026-10-12, finite range, not a proof): smallest working $K$ in range $28$; pairs found: 13 pairs.

Range searched: every $k\le 1000$, every $n$ with $2k\le n<k^2$ (complete in that range).
Pairs $(n,k)$ found that force $K\ge k$: 13 pairs, the last $(284,28)$.
Smallest $K$ that works in this range: $28$.

The method, the cross-checks and the code are in the numerical annex on erdos-1094 (the root), dated 2026-10-12.

How to read this page

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Explainer

Overview

The hole

Taking the bound

Supposing k exceeds K

A prime with the small remainder

The prime divides C(n, k)

The contradiction

Overview

The hole (h_sieve)

Taking the bound (s2)

Supposing k exceeds K (hKk)

A prime with the small remainder (s4)

The prime divides C(n, k) (hfact hzero hmodeq hdvd)

The contradiction (hle close)

How it is being proved

What is known

Work on it

Record