Open Proof Network

Problems / erdos-1094 / erdos-1094--h3--h1--h2--h2

Residue form, k and n odd, 3 ∤ k, no carry at 3

erdos-1094--h3--h1--h2--h2

open

Hole h_odd_rest of erdos-1094--h3--h1--h2 (split by 20261014T112100Z-t1014-z-partial.lean).

$\displaystyle \exists K_2\ \forall n,k\in\mathbb{N}:\quad K_2<k,\ \ 2k\le n<k^2,\ \ k \text{ odd},\ \ n \text{ odd},\ \ 3\nmid k,\ \ k \bmod 3 \le n \bmod 3 \implies \exists\,p\le k \text{ prime with } n \bmod p < k \bmod p$

For all large odd $k$ not divisible by $3$, every odd $n$ with $2k\le n<k^2$ whose residue mod $3$ is at least that of $k$ (so $p=3$ itself gives no carry) has a prime $p\le k$ whose residue of $n$ is smaller than its residue of $k$. Lean writes oddness as k % 2 = 1 and n % 2 = 1, $3\nmid k$ as 0 < k % 3, and $k^2$ as k * k. n % p is the least non-negative residue, so the condition reads $n \bmod p < k \bmod p$, a carry at the last base-$p$ digit when $k$ is subtracted from $n$ (hence $p \mid \binom{n}{k}$ by Kummer's theorem, though the statement names no binomial coefficient).

unverified · history

Lean
theorem erdos_1094__h3__h1__h2__h2 : ∃ (K₂ : ℕ),
  ∀ (n k : ℕ),
    K₂ < k →
      (2 : ℕ) * k ≤ n →
        n < k * k →
          k % (2 : ℕ) = (1 : ℕ) →
            n % (2 : ℕ) = (1 : ℕ) →
              (0 : ℕ) < k % (3 : ℕ) → k % (3 : ℕ) ≤ n % (3 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := 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 →

How it is being proved

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

Split into

Proved from these holes by 20261015T145200Z-t1015c-a11-partial.lean (merged in fbfc7a716a4d, attestation 000548).

  • h_k3_one → erdos-1094--h3--h1--h2--h2--h1 (open)

    $\displaystyle \exists K\ \forall n,k\in\mathbb{N}:\quad K<k,\ \ 2k\le n<k^2,\ \ k \text{ odd},\ \ n \text{ odd},\ \ k\equiv 1 \pmod 3 \implies \exists\,p\le k \text{ prime with } n \bmod p < k \bmod p$

  • h_k3_two → erdos-1094--h3--h1--h2--h2--h2 (open)

    $\displaystyle \exists K\ \forall n,k\in\mathbb{N}:\quad K<k,\ \ 2k\le n<k^2,\ \ k \text{ odd},\ \ n \text{ odd},\ \ k\equiv 2 \pmod 3,\ \ n\equiv 2 \pmod 3 \implies \exists\,p\le k \text{ prime with } n \bmod p < k \bmod p$

Read the partial proof (Lean)
theorem erdos_1094__h3__h1__h2__h2 : ∃ (K₂ : ℕ),
  ∀ (n k : ℕ),
    K₂ < k →
      (2 : ℕ) * k ≤ n →
        n < k * k →
          k % (2 : ℕ) = (1 : ℕ) →
            n % (2 : ℕ) = (1 : ℕ) →
              (0 : ℕ) < k % (3 : ℕ) → k % (3 : ℕ) ≤ n % (3 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := by
  have h_k3_one : ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k →
      k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (1 : ℕ) →
      ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := sorry
  have h_k3_two : ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k →
      k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (2 : ℕ) →
      n % (3 : ℕ) = (2 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := sorry
  obtain ⟨A, hA⟩ := h_k3_one
  obtain ⟨B, hB⟩ := h_k3_two
  refine ⟨max A B, ?_⟩
  intro n k hk h2k hnk hk2 hn2 h3 h3n
  have hk3 : k % 3 < 3 := Nat.mod_lt _ (by norm_num)
  have hn3 : n % 3 < 3 := Nat.mod_lt _ (by norm_num)
  rcases (show k % 3 = 1 ∨ k % 3 = 2 by omega) with h | h
  · exact hA n k (lt_of_le_of_lt (le_max_left _ _) hk) h2k hnk hk2 hn2 h
  · exact hB n k (lt_of_le_of_lt (le_max_right _ _) hk) h2k hnk hk2 hn2 h (by 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--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-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

Work on it

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 (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--h3--h1--h2--h2».Context

open scoped Nat

/-! Hole `h_odd_rest` 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__h1__h2__h2 : ∃ (K₂ : ℕ),
  ∀ (n k : ℕ),
    K₂ < k →
      (2 : ℕ) * k ≤ n →
        n < k * k →
          k % (2 : ℕ) = (1 : ℕ) →
            n % (2 : ℕ) = (1 : ℕ) →
              (0 : ℕ) < k % (3 : ℕ) → k % (3 : ℕ) ≤ n % (3 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := by
  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 →

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

$\displaystyle \exists K_2\ \forall n,k\in\mathbb{N}:\quad K_2<k,\ \ 2k\le n<k^2,\ \ k \text{ odd},\ \ n \text{ odd},\ \ 3\nmid k,\ \ k \bmod 3 \le n \bmod 3 \implies \exists\,p\le k \text{ prime with } n \bmod p < k \bmod p$

For all large odd $k$ not divisible by $3$, every odd $n$ with $2k\le n<k^2$ whose residue mod $3$ is at least that of $k$ (so $p=3$ itself gives no carry) has a prime $p\le k$ whose residue of $n$ is smaller than its residue of $k$. Lean writes oddness as k % 2 = 1 and n % 2 = 1, $3\nmid k$ as 0 < k % 3, and $k^2$ as k * k. n % p is the least non-negative residue, so the condition reads $n \bmod p < k \bmod p$, a carry at the last base-$p$ digit when $k$ is subtracted from $n$ (hence $p \mid \binom{n}{k}$ by Kummer's theorem, though the statement names no binomial coefficient).

Rendered from gloss/771cbe837cfe….md · Improve these words →

Rendered from targets/erdos-1094/nodes/erdos-1094--h3--h1--h2--h2/Statement.lean. Origin: compiler-derived.

Depends on (2), each in words

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

  • erdos-1094--h3--h1--h2--h2--h1: $\exists K\ \forall n,k\in\mathbb{N}:\quad K<k,\ \ 2k\le n<k^2,\ \ k \text{ odd},\ \ n \text{ odd},\ \ k\equiv 1 \pmod 3 \implies \exists\,p\le k \text{ prime with } n \bmod p < k \bmod p$
  • erdos-1094--h3--h1--h2--h2--h2: $\exists K\ \forall n,k\in\mathbb{N}:\quad K<k,\ \ 2k\le n<k^2,\ \ k \text{ odd},\ \ n \text{ odd},\ \ k\equiv 2 \pmod 3,\ \ n\equiv 2 \pmod 3 \implies \exists\,p\le k \text{ prime with } n \bmod p < k \bmod p$
Attestation: none of a proof

No proof of this statement has been merged. A partial assembly merged under attestations/000548.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--h1--h2--h2/Witness.lean, sha256 1aa8a1c1b9f3.
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 (2)

2 recorded. Routes refuted: none. Failure classes: route-dead-ends 1.

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 t1015c-a11; merged in fbfc7a716a4d under attestations/000548.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--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-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 20261015T145200Z-t1015c-a11-partial.lean, extracted by the gate at 067497e74a8b 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.

  1. h_k3_one hole ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (1 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p hole

    Leaves ∃ (K₂ : ℕ), ∀ (n k : ℕ), K₂ < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → (0 : ℕ) < k % (3 : ℕ) → k % (3 : ℕ) ≤ n % (3 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p, with h_k3_one : ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (1 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p.

    Uses library Nat.Prime Nat.Prime p means that p is a prime number, that is, a natural number at least 2 whose only divisors are p and 1. .

    Left open as a hole; its statement is erdos-1094--h3--h1--h2--h2--h1 (open).

    Lean, lines 17–19
      have h_k3_one : ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k →
      k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (1 : ℕ) →
      ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := sorry

    targets/erdos-1094/nodes/erdos-1094--h3--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-partial.lean, lines 17–19 at 6ba7fea98468 ↗

  2. h_k3_two hole ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (2 : ℕ) → n % (3 : ℕ) = (2 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p hole

    Leaves ∃ (K₂ : ℕ), ∀ (n k : ℕ), K₂ < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → (0 : ℕ) < k % (3 : ℕ) → k % (3 : ℕ) ≤ n % (3 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p, with h_k3_two : ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (2 : ℕ) → n % (3 : ℕ) = (2 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p.

    Uses library Nat.Prime Nat.Prime p means that p is a prime number, that is, a natural number at least 2 whose only divisors are p and 1. .

    Left open as a hole; its statement is erdos-1094--h3--h1--h2--h2--h2 (open).

    Lean, lines 20–22
      have h_k3_two : ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k →
      k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (2 : ℕ) →
      n % (3 : ℕ) = (2 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := sorry

    targets/erdos-1094/nodes/erdos-1094--h3--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-partial.lean, lines 20–22 at 6ba7fea98468 ↗

  3. s3 obtain ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (1 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p

    Leaves ∃ (K₂ : ℕ), ∀ (n k : ℕ), K₂ < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → (0 : ℕ) < k % (3 : ℕ) → k % (3 : ℕ) ≤ n % (3 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p, with A : ℕ, hA : ∀ (n k : ℕ), A < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (1 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p.

    Lean, line 23
      obtain ⟨A, hA⟩ := h_k3_one

    targets/erdos-1094/nodes/erdos-1094--h3--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-partial.lean, line 23 at 6ba7fea98468 ↗

  4. s4 obtain ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (2 : ℕ) → n % (3 : ℕ) = (2 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p

    Leaves ∃ (K₂ : ℕ), ∀ (n k : ℕ), K₂ < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → (0 : ℕ) < k % (3 : ℕ) → k % (3 : ℕ) ≤ n % (3 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p, with B : ℕ, hB : ∀ (n k : ℕ), B < k → (2 : ℕ) * k ≤ n → n < k * k → k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (2 : ℕ) → n % (3 : ℕ) = (2 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p.

    Lean, line 24
      obtain ⟨B, hB⟩ := h_k3_two

    targets/erdos-1094/nodes/erdos-1094--h3--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-partial.lean, line 24 at 6ba7fea98468 ↗

  5. hk3 have k % (3 : ℕ) < (3 : ℕ)

    Leaves ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p, with hk3 : k % (3 : ℕ) < (3 : ℕ).

    Uses library Nat.mod_lt.

    Lean, line 27
      have hk3 : k % 3 < 3 := Nat.mod_lt _ (by norm_num)

    targets/erdos-1094/nodes/erdos-1094--h3--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-partial.lean, line 27 at 6ba7fea98468 ↗

  6. hn3 have n % (3 : ℕ) < (3 : ℕ)

    Leaves ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p, with hn3 : n % (3 : ℕ) < (3 : ℕ).

    Uses library Nat.mod_lt.

    Lean, line 28
      have hn3 : n % 3 < 3 := Nat.mod_lt _ (by norm_num)

    targets/erdos-1094/nodes/erdos-1094--h3--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-partial.lean, line 28 at 6ba7fea98468 ↗

  7. s7 case

    Leaves ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p, with h : k % (3 : ℕ) = (1 : ℕ).

    Uses library lt_of_le_of_lt, le_max_left.

    Lean, line 30
      · exact hA n k (lt_of_le_of_lt (le_max_left _ _) hk) h2k hnk hk2 hn2 h

    targets/erdos-1094/nodes/erdos-1094--h3--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-partial.lean, line 30 at 6ba7fea98468 ↗

  8. s8 case

    Leaves ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p, with h : k % (3 : ℕ) = (2 : ℕ).

    Uses library lt_of_le_of_lt, le_max_right.

    Lean, line 31
      · exact hB n k (lt_of_le_of_lt (le_max_right _ _) hk) h2k hnk hk2 hn2 h (by omega)

    targets/erdos-1094/nodes/erdos-1094--h3--h1--h2--h2/attempts/20261015T145200Z-t1015c-a11-partial.lean, line 31 at 6ba7fea98468 ↗

import Mathlib
import Nodes.«erdos-1094--h3--h1--h2--h2».Context

open scoped Nat

/-! Hole `h_odd_rest` 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__h1__h2__h2 : ∃ (K₂ : ℕ),
  ∀ (n k : ℕ),
    K₂ < k →
      (2 : ℕ) * k ≤ n →
        n < k * k →
          k % (2 : ℕ) = (1 : ℕ) →
            n % (2 : ℕ) = (1 : ℕ) →
              (0 : ℕ) < k % (3 : ℕ) → k % (3 : ℕ) ≤ n % (3 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := by
  have h_k3_one : ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k →
      k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (1 : ℕ) →
      ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := sorry
  have h_k3_two : ∃ (K : ℕ), ∀ (n k : ℕ), K < k → (2 : ℕ) * k ≤ n → n < k * k →
      k % (2 : ℕ) = (1 : ℕ) → n % (2 : ℕ) = (1 : ℕ) → k % (3 : ℕ) = (2 : ℕ) →
      n % (3 : ℕ) = (2 : ℕ) → ∃ (p : ℕ), Nat.Prime p ∧ p ≤ k ∧ n % p < k % p := sorry
  obtain ⟨A, hA⟩ := h_k3_one
  obtain ⟨B, hB⟩ := h_k3_two
  refine ⟨max A B, ?_⟩
  intro n k hk h2k hnk hk2 hn2 h3 h3n
  have hk3 : k % 3 < 3 := Nat.mod_lt _ (by norm_num)
  have hn3 : n % 3 < 3 := Nat.mod_lt _ (by norm_num)
  rcases (show k % 3 = 1 ∨ k % 3 = 2 by omega) with h | h
  · exact hA n k (lt_of_le_of_lt (le_max_left _ _) hk) h2k hnk hk2 hn2 h
  · exact hB n k (lt_of_le_of_lt (le_max_right _ _) hk) h2k hnk hk2 hn2 h (by omega)
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).

How to read a statement’s page →