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Problems / erdos-1094 / erdos-1094--h3--h3

Granville–Ramaré: a carry at some prime p in (k/2, 11k/20)

erdos-1094--h3--h3

open

Hole h_gr of erdos-1094--h3 (split by 20261015T125200Z-t1015b-g-partial.lean).

$\displaystyle \forall C\in\mathbb{N}\ \exists K_2\ \forall n,k\in\mathbb{N}:\quad K_2<k,\ \ k^2\le C\cdot n\cdot\lfloor\log_2 k\rfloor,\ \ n<k^2 \implies \exists\,p \text{ prime},\ \tfrac{k}{2}<p<\tfrac{11k}{20},\ \ n \bmod p < k \bmod p$

For every constant $C$ there is a threshold $K_2$ such that for all $k>K_2$ and all $n<k^2$ with $k^2\le C\,n\lfloor\log_2 k\rfloor$ (the range $n\gg k^2/\log k$ that the ELS bound leaves), some prime $p$ with $k<2p$ and $20p<11k$ has $n \bmod p<k \bmod p$, a carry at the last base-$p$ digit. Everything is in $\mathbb{N}$: Nat.log 2 k is $\lfloor\log_2 k\rfloor$ (and $0$ for $k\le 1$), a natural number, so the right-hand side is a product of naturals, and n % p is the least non-negative residue.

unverified · history

Lean
theorem erdos_1094__h3__h3 : ∀ (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 := 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 20261015T132200Z-t1015b-g-partial.lean (merged in 0f3cc6356058, attestation 000542).

  • h_cheb → erdos-1094--h3--h3--h1 (open)

    $\displaystyle \exists K_3\in\mathbb{N}\ \forall k\in\mathbb{N}:\quad K_3<k \implies \frac{k}{100} \le \sum_{\substack{p \text{ prime}\\ k/2<p<11k/20}} \log p$

  • h_sander → erdos-1094--h3--h3--h2 (open)

    $\displaystyle \forall C\in\mathbb{N},\ \varepsilon>0\ \exists K_4\ \forall n,k\in\mathbb{N}:\quad K_4<k,\ \ k^2\le C\cdot n\cdot\lfloor\log_2 k\rfloor,\ \ n<k^2 \implies \Bigl|\sum_{\substack{p \text{ prime}\\ k/2<p<11k/20}} \Bigl(\Bigl\{\frac{n}{p}\Bigr\}-\frac12\Bigr)\log p\Bigr| \le \varepsilon k$

Read the partial proof (Lean)
theorem erdos_1094__h3__h3 : ∀ (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 := by
  -- annex: 88f9e9a7ddffdc9d2a479880cf6eaa066f94ea7f17936fe431b479452d6c05bd
  -- Second level of Granville–Ramaré, Mathematika 43 (1996) 73–107, Theorem 8, §5e
  -- (preprint pp. 24–25), the analytic step. Suppose no prime p in (k/2, 11k/20) has a
  -- last-digit carry; then {n/p} ≥ k/p − 1 > 9/11 for each such p (GR read the same
  -- no-carry condition as {m/p} < 1/5 with m = n − k), so Σ ({n/p} − 1/2) log p is
  -- ≥ (7/22) Σ log p, which h_cheb makes ≫ k and h_sander makes o(k).
  -- h_cheb: the "≫ k" of p. 25, used and not proved by GR: Σ_{k/2<p<11k/20} log p ≫ k,
  -- the prime number theorem on (x, 1.1x) (θ(1.1x) − θ(x) ~ x/10 with x = k/2).
  have h_cheb : ∃ K₃ : ℕ, ∀ k : ℕ, K₃ < k →
      (k : ℝ) / 100 ≤ ∑ p ∈ (Finset.range k).filter (fun p => p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k),
        Real.log p := by
    sorry
  -- h_sander: GR's (5.1) (p. 18; the J = 1 case of their Lemma 5.1, taken from Sander [Sa2],
  -- J. reine angew. Math. 430) on the interval (k/2, 11k/20), with its error terms o(k) on
  -- this node's range (k² ≪ n log k, n < k²). Applied at x = n rather than GR's n − k: (5.1)
  -- is uniform in x. Written with {t} − 1/2 in place of GR's ψ(t) (ψ is 0 at integers); the two
  -- differ only at primes p | n, at most two in (k/2, k] when n < k² and k > 8, so O(log k).
  have h_sander : ∀ (C : ℕ) (ε : ℝ), 0 < ε → ∃ K₄ : ℕ, ∀ n k : ℕ, K₄ < k →
      k * k ≤ C * (n * Nat.log 2 k) → n < k * k →
      |∑ p ∈ (Finset.range k).filter (fun p => p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k),
        (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p| ≤ ε * k := by
    sorry
  intro C
  obtain ⟨K₃, hc⟩ := h_cheb
  obtain ⟨K₄, hs⟩ := h_sander C (1 / 1000) (by norm_num)
  refine ⟨max K₃ K₄, fun n k hk hlow hup => ?_⟩
  have hk3 : K₃ < k := lt_of_le_of_lt (le_max_left _ _) hk
  have hk4 : K₄ < k := lt_of_le_of_lt (le_max_right _ _) hk
  by_contra hno
  push_neg at hno
  set S := (Finset.range k).filter (fun p => p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k) with hS
  have hterm : ∀ p ∈ S, (7 / 22 : ℝ) * Real.log p ≤ (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p := by
    intro p hp
    rw [hS, Finset.mem_filter] at hp
    obtain ⟨-, hpr, h2p, h20⟩ := hp
    have hmod := hno p hpr h2p h20
    have hpk : p ≤ k := by omega
    have hkp : k % p = k - p := by
      rw [Nat.mod_eq_sub_mod hpk, Nat.mod_eq_of_lt (by omega)]
    have hp0 : (0 : ℝ) < p := by exact_mod_cast hpr.pos
    have hfr : Int.fract ((n : ℝ) / p) = ((n % p : ℕ) : ℝ) / p := Int.fract_div_natCast_eq_div_natCast_mod
    have hge : ((k - p : ℕ) : ℝ) ≤ ((n % p : ℕ) : ℝ) := by exact_mod_cast (hkp ▸ hmod)
    have hsub : ((k - p : ℕ) : ℝ) = (k : ℝ) - p := by
      rw [Nat.cast_sub hpk]
    have h20r : (20 : ℝ) * p < 11 * k := by exact_mod_cast h20
    have hfrac : (9 / 11 : ℝ) ≤ Int.fract ((n : ℝ) / p) := by
      rw [hfr, le_div_iff₀ hp0]
      linarith
    have hlog : 0 ≤ Real.log p := Real.log_natCast_nonneg p
    nlinarith
  have hsum : (7 / 22 : ℝ) * ∑ p ∈ S, Real.log p ≤
      ∑ p ∈ S, (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p := by
    rw [Finset.mul_sum]
    exact Finset.sum_le_sum hterm
  have hc' := hc k hk3
  have hs' := hs n k hk4 hlow hup
  have habs := le_abs_self (∑ p ∈ S, (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p)
  have hkpos : (0 : ℝ) < k := by exact_mod_cast (show 0 < k by omega)
  linarith

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--h3/attempts/20261015T132200Z-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

Work on it

Plans

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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--h3».Context

open scoped Nat

/-! Hole `h_gr` 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__h3 : ∀ (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 := 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

$\displaystyle \forall C\in\mathbb{N}\ \exists K_2\ \forall n,k\in\mathbb{N}:\quad K_2<k,\ \ k^2\le C\cdot n\cdot\lfloor\log_2 k\rfloor,\ \ n<k^2 \implies \exists\,p \text{ prime},\ \tfrac{k}{2}<p<\tfrac{11k}{20},\ \ n \bmod p < k \bmod p$

For every constant $C$ there is a threshold $K_2$ such that for all $k>K_2$ and all $n<k^2$ with $k^2\le C\,n\lfloor\log_2 k\rfloor$ (the range $n\gg k^2/\log k$ that the ELS bound leaves), some prime $p$ with $k<2p$ and $20p<11k$ has $n \bmod p<k \bmod p$, a carry at the last base-$p$ digit. Everything is in $\mathbb{N}$: Nat.log 2 k is $\lfloor\log_2 k\rfloor$ (and $0$ for $k\le 1$), a natural number, so the right-hand side is a product of naturals, and n % p is the least non-negative residue.

Rendered from gloss/db10fba82234….md · Improve these words →

Rendered from targets/erdos-1094/nodes/erdos-1094--h3--h3/Statement.lean. Origin: skeleton-hole.

Depends on (2), each in words

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

  • erdos-1094--h3--h3--h1: $\exists K_3\in\mathbb{N}\ \forall k\in\mathbb{N}:\quad K_3<k \implies \frac{k}{100} \le \sum_{\substack{p \text{ prime}\\ k/2<p<11k/20}} \log p$
  • erdos-1094--h3--h3--h2: $\forall C\in\mathbb{N},\ \varepsilon>0\ \exists K_4\ \forall n,k\in\mathbb{N}:\quad K_4<k,\ \ k^2\le C\cdot n\cdot\lfloor\log_2 k\rfloor,\ \ n<k^2 \implies \Bigl|\sum_{\substack{p \text{ prime}\\ k/2<p<11k/20}} \Bigl(\Bigl\{\frac{n}{p}\Bigr\}-\frac12\Bigr)\log p\Bigr| \le \varepsilon k$
Attestation: none of a proof

No proof of this statement has been merged. A partial assembly merged under attestations/000542.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--h3/Witness.lean, sha256 0dbf380a6878.
import Mathlib

open scoped Nat

theorem witness : ∃ (C : ℕ), True := ⟨0, 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 (1)

1 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 t1015b-g; merged in 0f3cc6356058 under attestations/000542.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--h3/attempts/20261015T132200Z-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 20261015T132200Z-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.

  1. h_cheb hole ∃ (K₃ : ℕ), ∀ (k : ℕ), K₃ < k → (↑k : ℝ) / (100 : ℝ) ≤ ∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, Real.log (↑p : ℝ) hole

    Leaves ∀ (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, with h_cheb : ∃ (K₃ : ℕ), ∀ (k : ℕ), K₃ < k → (↑k : ℝ) / (100 : ℝ) ≤ ∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, Real.log (↑p : ℝ).

    Uses library Finset.range range n is the set of natural numbers less than n. , Real.log The real logarithm function, equal to the inverse of the exponential for x > 0, to log |x| for x < 0, and to 0 for 0. .

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

    Lean, lines 23–26
      have h_cheb : ∃ K₃ : ℕ, ∀ k : ℕ, K₃ < k →
      (k : ℝ) / 100 ≤ ∑ p ∈ (Finset.range k).filter (fun p => p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k),
        Real.log p := by
    sorry

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, lines 23–26 at 6ba7fea98468 ↗

  2. h_sander hole ∀ (C : ℕ) (ε : ℝ), (0 : ℝ) < ε → ∃ (K₄ : ℕ), ∀ (n k : ℕ), K₄ < k → k * k ≤ C * (n * Nat.log (2 : ℕ) k) → n < k * k → |∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)| ≤ ε * (↑k : ℝ) hole

    Leaves ∀ (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, with h_sander : ∀ (C : ℕ) (ε : ℝ), (0 : ℝ) < ε → ∃ (K₄ : ℕ), ∀ (n k : ℕ), K₄ < k → k * k ≤ C * (n * Nat.log (2 : ℕ) k) → n < k * k → |∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)| ≤ ε * (↑k : ℝ).

    Uses library Nat.log log b n, is the logarithm of natural number n in base b. , Finset.range range n is the set of natural numbers less than n. , Int.fract Int.fract a the fractional part of a, is a minus its floor. , Real.log The real logarithm function, equal to the inverse of the exponential for x > 0, to log |x| for x < 0, and to 0 for 0. .

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

    Lean, lines 32–36
      have h_sander : ∀ (C : ℕ) (ε : ℝ), 0 < ε → ∃ K₄ : ℕ, ∀ n k : ℕ, K₄ < k →
      k * k ≤ C * (n * Nat.log 2 k) → n < k * k →
      |∑ p ∈ (Finset.range k).filter (fun p => p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k),
        (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p| ≤ ε * k := by
    sorry

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, lines 32–36 at 6ba7fea98468 ↗

  3. s3 obtain ∃ (K₃ : ℕ), ∀ (k : ℕ), K₃ < k → (↑k : ℝ) / (100 : ℝ) ≤ ∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, Real.log (↑p : ℝ)

    Leaves ∃ (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, with K₃ : ℕ, hc : ∀ (k : ℕ), K₃ < k → (↑k : ℝ) / (100 : ℝ) ≤ ∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, Real.log (↑p : ℝ).

    Lean, line 38
      obtain ⟨K₃, hc⟩ := h_cheb

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 38 at 6ba7fea98468 ↗

  4. s4 obtain ∃ (K₄ : ℕ), ∀ (n k : ℕ), K₄ < k → k * k ≤ C * (n * Nat.log (2 : ℕ) k) → n < k * k → |∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)| ≤ (1 / 1000 : ℝ) * (↑k : ℝ)

    Leaves ∃ (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, with K₄ : ℕ, hs : ∀ (n k : ℕ), K₄ < k → k * k ≤ C * (n * Nat.log (2 : ℕ) k) → n < k * k → |∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)| ≤ (1 / 1000 : ℝ) * (↑k : ℝ).

    Lean, line 39
      obtain ⟨K₄, hs⟩ := h_sander C (1 / 1000) (by norm_num)

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 39 at 6ba7fea98468 ↗

  5. hk3 have K₃ < k

    Leaves ∃ (p : ℕ), Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k ∧ n % p < k % p, with hk3 : K₃ < k.

    Uses library lt_of_le_of_lt, le_max_left.

    Lean, line 41
      have hk3 : K₃ < k := lt_of_le_of_lt (le_max_left _ _) hk

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 41 at 6ba7fea98468 ↗

  6. hk4 have K₄ < k

    Leaves ∃ (p : ℕ), Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k ∧ n % p < k % p, with hk4 : K₄ < k.

    Uses library lt_of_le_of_lt, le_max_right.

    Lean, line 42
      have hk4 : K₄ < k := lt_of_le_of_lt (le_max_right _ _) hk

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 42 at 6ba7fea98468 ↗

  7. hterm have ∀ p ∈ S, (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)

    Leaves False, with hterm : ∀ p ∈ S, (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ).

    Uses library Real.log The real logarithm function, equal to the inverse of the exponential for x > 0, to log |x| for x < 0, and to 0 for 0. , Int.fract Int.fract a the fractional part of a, is a minus its floor. , Finset.mem_filter.

    Lean, lines 46–64
      have hterm : ∀ p ∈ S, (7 / 22 : ℝ) * Real.log p ≤ (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p := by
    intro p hp
    rw [hS, Finset.mem_filter] at hp
    obtain ⟨-, hpr, h2p, h20⟩ := hp
    have hmod := hno p hpr h2p h20
    have hpk : p ≤ k := by omega
    have hkp : k % p = k - p := by
      rw [Nat.mod_eq_sub_mod hpk, Nat.mod_eq_of_lt (by omega)]
    have hp0 : (0 : ℝ) < p := by exact_mod_cast hpr.pos
    have hfr : Int.fract ((n : ℝ) / p) = ((n % p : ℕ) : ℝ) / p := Int.fract_div_natCast_eq_div_natCast_mod
    have hge : ((k - p : ℕ) : ℝ) ≤ ((n % p : ℕ) : ℝ) := by exact_mod_cast (hkp ▸ hmod)
    have hsub : ((k - p : ℕ) : ℝ) = (k : ℝ) - p := by
      rw [Nat.cast_sub hpk]
    have h20r : (20 : ℝ) * p < 11 * k := by exact_mod_cast h20
    have hfrac : (9 / 11 : ℝ) ≤ Int.fract ((n : ℝ) / p) := by
      rw [hfr, le_div_iff₀ hp0]
      linarith
    have hlog : 0 ≤ Real.log p := Real.log_natCast_nonneg p
    nlinarith

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, lines 46–64 at 6ba7fea98468 ↗

    1. hterm.s1 obtain p ∈ Finset.range k ∧ Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hpr : Nat.Prime p, h2p : k < (2 : ℕ) * p, h20 : (20 : ℕ) * p < (11 : ℕ) * k.

      Lean, line 49
          obtain ⟨-, hpr, h2p, h20⟩ := hp

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 49 at 6ba7fea98468 ↗

    2. hterm.hmod have k % p ≤ n % p

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hmod : k % p ≤ n % p.

      Lean, line 50
          have hmod := hno p hpr h2p h20

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 50 at 6ba7fea98468 ↗

    3. hterm.hpk have p ≤ k routine: omega

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hpk : p ≤ k.

      Lean, line 51
          have hpk : p ≤ k := by omega

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 51 at 6ba7fea98468 ↗

    4. hterm.hkp have k % p = k - p

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hkp : k % p = k - p.

      Uses library Nat.mod_eq_sub_mod, Nat.mod_eq_of_lt.

      Lean, lines 52–53
          have hkp : k % p = k - p := by
      rw [Nat.mod_eq_sub_mod hpk, Nat.mod_eq_of_lt (by omega)]

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, lines 52–53 at 6ba7fea98468 ↗

    5. hterm.hp0 have (0 : ℝ) < (↑p : ℝ)

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hp0 : (0 : ℝ) < (↑p : ℝ).

      Lean, line 54
          have hp0 : (0 : ℝ) < p := by exact_mod_cast hpr.pos

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 54 at 6ba7fea98468 ↗

    6. hterm.hfr have Int.fract ((↑n : ℝ) / (↑p : ℝ)) = (↑(n % p) : ℝ) / (↑p : ℝ)

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hfr : Int.fract ((↑n : ℝ) / (↑p : ℝ)) = (↑(n % p) : ℝ) / (↑p : ℝ).

      Uses library Int.fract Int.fract a the fractional part of a, is a minus its floor. , Int.fract_div_natCast_eq_div_natCast_mod.

      Lean, line 55
          have hfr : Int.fract ((n : ℝ) / p) = ((n % p : ℕ) : ℝ) / p := Int.fract_div_natCast_eq_div_natCast_mod

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 55 at 6ba7fea98468 ↗

    7. hterm.hge have (↑(k - p) : ℝ) ≤ (↑(n % p) : ℝ)

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hge : (↑(k - p) : ℝ) ≤ (↑(n % p) : ℝ).

      Lean, line 56
          have hge : ((k - p : ℕ) : ℝ) ≤ ((n % p : ℕ) : ℝ) := by exact_mod_cast (hkp ▸ hmod)

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 56 at 6ba7fea98468 ↗

    8. hterm.hsub have (↑(k - p) : ℝ) = (↑k : ℝ) - (↑p : ℝ)

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hsub : (↑(k - p) : ℝ) = (↑k : ℝ) - (↑p : ℝ).

      Uses library Nat.cast_sub.

      Lean, lines 57–58
          have hsub : ((k - p : ℕ) : ℝ) = (k : ℝ) - p := by
      rw [Nat.cast_sub hpk]

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, lines 57–58 at 6ba7fea98468 ↗

    9. hterm.h20r have (20 : ℝ) * (↑p : ℝ) < (11 : ℝ) * (↑k : ℝ)

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with h20r : (20 : ℝ) * (↑p : ℝ) < (11 : ℝ) * (↑k : ℝ).

      Lean, line 59
          have h20r : (20 : ℝ) * p < 11 * k := by exact_mod_cast h20

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 59 at 6ba7fea98468 ↗

    10. hterm.hfrac have (9 / 11 : ℝ) ≤ Int.fract ((↑n : ℝ) / (↑p : ℝ))

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hfrac : (9 / 11 : ℝ) ≤ Int.fract ((↑n : ℝ) / (↑p : ℝ)).

      Uses library Int.fract Int.fract a the fractional part of a, is a minus its floor. , le_div_iff₀ See le_div_iff₀' for a version with multiplication on the other side. .

      Lean, lines 60–62
          have hfrac : (9 / 11 : ℝ) ≤ Int.fract ((n : ℝ) / p) := by
      rw [hfr, le_div_iff₀ hp0]
      linarith

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, lines 60–62 at 6ba7fea98468 ↗

    11. hterm.hlog have (0 : ℝ) ≤ Real.log (↑p : ℝ)

      Leaves (7 / 22 : ℝ) * Real.log (↑p : ℝ) ≤ (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ), with hlog : (0 : ℝ) ≤ Real.log (↑p : ℝ).

      Uses library Real.log The real logarithm function, equal to the inverse of the exponential for x > 0, to log |x| for x < 0, and to 0 for 0. , Real.log_natCast_nonneg.

      Lean, line 63
          have hlog : 0 ≤ Real.log p := Real.log_natCast_nonneg p

      targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 63 at 6ba7fea98468 ↗

  8. hsum have (7 / 22 : ℝ) * ∑ p ∈ S, Real.log (↑p : ℝ) ≤ ∑ p ∈ S, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)

    Leaves False, with hsum : (7 / 22 : ℝ) * ∑ p ∈ S, Real.log (↑p : ℝ) ≤ ∑ p ∈ S, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ).

    Uses library Real.log The real logarithm function, equal to the inverse of the exponential for x > 0, to log |x| for x < 0, and to 0 for 0. , Int.fract Int.fract a the fractional part of a, is a minus its floor. , Finset.mul_sum, Finset.sum_le_sum In an ordered additive commutative monoid, if each summand f i of one finite sum is less than or equal to the corresponding summand g i of another finite sum, then ∑ i ∈ s, f i ≤ ∑ i ∈ s, g i. .

    Lean, lines 65–68
      have hsum : (7 / 22 : ℝ) * ∑ p ∈ S, Real.log p ≤
      ∑ p ∈ S, (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p := by
    rw [Finset.mul_sum]
    exact Finset.sum_le_sum hterm

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, lines 65–68 at 6ba7fea98468 ↗

  9. hc' have (↑k : ℝ) / (100 : ℝ) ≤ ∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, Real.log (↑p : ℝ)

    Leaves False, with hc' : (↑k : ℝ) / (100 : ℝ) ≤ ∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, Real.log (↑p : ℝ).

    Lean, line 69
      have hc' := hc k hk3

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 69 at 6ba7fea98468 ↗

  10. hs' have |∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)| ≤ (1 / 1000 : ℝ) * (↑k : ℝ)

    Leaves False, with hs' : |∑ p ∈ Finset.range k with Nat.Prime p ∧ k < (2 : ℕ) * p ∧ (20 : ℕ) * p < (11 : ℕ) * k, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)| ≤ (1 / 1000 : ℝ) * (↑k : ℝ).

    Lean, line 70
      have hs' := hs n k hk4 hlow hup

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 70 at 6ba7fea98468 ↗

  11. habs have ∑ p ∈ S, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ) ≤ |∑ p ∈ S, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)|

    Leaves False, with habs : ∑ p ∈ S, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ) ≤ |∑ p ∈ S, (Int.fract ((↑n : ℝ) / (↑p : ℝ)) - (1 / 2 : ℝ)) * Real.log (↑p : ℝ)|.

    Uses library le_abs_self, Int.fract Int.fract a the fractional part of a, is a minus its floor. , Real.log The real logarithm function, equal to the inverse of the exponential for x > 0, to log |x| for x < 0, and to 0 for 0. .

    Lean, line 71
      have habs := le_abs_self (∑ p ∈ S, (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p)

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 71 at 6ba7fea98468 ↗

  12. hkpos have (0 : ℝ) < (↑k : ℝ)

    Leaves False, with hkpos : (0 : ℝ) < (↑k : ℝ).

    Lean, line 72
      have hkpos : (0 : ℝ) < k := by exact_mod_cast (show 0 < k by omega)

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 72 at 6ba7fea98468 ↗

  13. close term False

    Closes the goal.

    Lean, line 73
      linarith

    targets/erdos-1094/nodes/erdos-1094--h3--h3/attempts/20261015T132200Z-t1015b-g-partial.lean, line 73 at 6ba7fea98468 ↗

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

open scoped Nat

/-! Hole `h_gr` 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__h3 : ∀ (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 := by
  -- annex: 88f9e9a7ddffdc9d2a479880cf6eaa066f94ea7f17936fe431b479452d6c05bd
  -- Second level of Granville–Ramaré, Mathematika 43 (1996) 73–107, Theorem 8, §5e
  -- (preprint pp. 24–25), the analytic step. Suppose no prime p in (k/2, 11k/20) has a
  -- last-digit carry; then {n/p} ≥ k/p − 1 > 9/11 for each such p (GR read the same
  -- no-carry condition as {m/p} < 1/5 with m = n − k), so Σ ({n/p} − 1/2) log p is
  -- ≥ (7/22) Σ log p, which h_cheb makes ≫ k and h_sander makes o(k).
  -- h_cheb: the "≫ k" of p. 25, used and not proved by GR: Σ_{k/2<p<11k/20} log p ≫ k,
  -- the prime number theorem on (x, 1.1x) (θ(1.1x) − θ(x) ~ x/10 with x = k/2).
  have h_cheb : ∃ K₃ : ℕ, ∀ k : ℕ, K₃ < k →
      (k : ℝ) / 100 ≤ ∑ p ∈ (Finset.range k).filter (fun p => p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k),
        Real.log p := by
    sorry
  -- h_sander: GR's (5.1) (p. 18; the J = 1 case of their Lemma 5.1, taken from Sander [Sa2],
  -- J. reine angew. Math. 430) on the interval (k/2, 11k/20), with its error terms o(k) on
  -- this node's range (k² ≪ n log k, n < k²). Applied at x = n rather than GR's n − k: (5.1)
  -- is uniform in x. Written with {t} − 1/2 in place of GR's ψ(t) (ψ is 0 at integers); the two
  -- differ only at primes p | n, at most two in (k/2, k] when n < k² and k > 8, so O(log k).
  have h_sander : ∀ (C : ℕ) (ε : ℝ), 0 < ε → ∃ K₄ : ℕ, ∀ n k : ℕ, K₄ < k →
      k * k ≤ C * (n * Nat.log 2 k) → n < k * k →
      |∑ p ∈ (Finset.range k).filter (fun p => p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k),
        (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p| ≤ ε * k := by
    sorry
  intro C
  obtain ⟨K₃, hc⟩ := h_cheb
  obtain ⟨K₄, hs⟩ := h_sander C (1 / 1000) (by norm_num)
  refine ⟨max K₃ K₄, fun n k hk hlow hup => ?_⟩
  have hk3 : K₃ < k := lt_of_le_of_lt (le_max_left _ _) hk
  have hk4 : K₄ < k := lt_of_le_of_lt (le_max_right _ _) hk
  by_contra hno
  push_neg at hno
  set S := (Finset.range k).filter (fun p => p.Prime ∧ k < 2 * p ∧ 20 * p < 11 * k) with hS
  have hterm : ∀ p ∈ S, (7 / 22 : ℝ) * Real.log p ≤ (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p := by
    intro p hp
    rw [hS, Finset.mem_filter] at hp
    obtain ⟨-, hpr, h2p, h20⟩ := hp
    have hmod := hno p hpr h2p h20
    have hpk : p ≤ k := by omega
    have hkp : k % p = k - p := by
      rw [Nat.mod_eq_sub_mod hpk, Nat.mod_eq_of_lt (by omega)]
    have hp0 : (0 : ℝ) < p := by exact_mod_cast hpr.pos
    have hfr : Int.fract ((n : ℝ) / p) = ((n % p : ℕ) : ℝ) / p := Int.fract_div_natCast_eq_div_natCast_mod
    have hge : ((k - p : ℕ) : ℝ) ≤ ((n % p : ℕ) : ℝ) := by exact_mod_cast (hkp ▸ hmod)
    have hsub : ((k - p : ℕ) : ℝ) = (k : ℝ) - p := by
      rw [Nat.cast_sub hpk]
    have h20r : (20 : ℝ) * p < 11 * k := by exact_mod_cast h20
    have hfrac : (9 / 11 : ℝ) ≤ Int.fract ((n : ℝ) / p) := by
      rw [hfr, le_div_iff₀ hp0]
      linarith
    have hlog : 0 ≤ Real.log p := Real.log_natCast_nonneg p
    nlinarith
  have hsum : (7 / 22 : ℝ) * ∑ p ∈ S, Real.log p ≤
      ∑ p ∈ S, (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p := by
    rw [Finset.mul_sum]
    exact Finset.sum_le_sum hterm
  have hc' := hc k hk3
  have hs' := hs n k hk4 hlow hup
  have habs := le_abs_self (∑ p ∈ S, (Int.fract ((n : ℝ) / p) - 1 / 2) * Real.log p)
  have hkpos : (0 : ℝ) < k := by exact_mod_cast (show 0 < k by omega)
  linarith
Annexes (1), 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, 2026-10-15T13:21:38Z, licence CC-BY-4.0. Rendered from targets/erdos-1094/nodes/erdos-1094--h3--h3/annex/88f9e9a7ddffdc9d2a479880cf6eaa066f94ea7f17936fe431b479452d6c05bd.md.

The analytic step of Granville–Ramaré's Theorem 8 (§5e, pp. 24–25 of their preprint), split along its own seams: a prime-sum lower bound and the equidistribution bound (5.1).

Source. A. Granville and O. Ramaré, Mathematika 43 (1996) 73–107; page numbers are the authors' preprint's. Read by t1015b-g and t1015b-h on 2026-10-15. This node is the hole h_gr of the skeleton on erdos-1094--h3 (annex there).

The statement. For every C there is K such that for k > K, k² ≤ C·n·log₂k and n < k², some prime p with k/2 < p < 11k/20 has n mod p < k mod p.

GR's argument (p. 25), and the form used here. Suppose no such p. For such p, k mod p = k − p, so n mod p ≥ k − p, i.e. the fractional part {n/p} ≥ k/p − 1 > 20/11 − 1 = 9/11. (GR work with m = n − k and get {m/p} < 1/5 for every such p; the condition is the same no-carry condition read from the other summand, and using n avoids natural subtraction. This is our deviation in form, not in substance.) Hence each term ({n/p} − 1/2) log p is at least (7/22) log p, and

Σ_{k/2<p<11k/20} ({n/p} − 1/2) log p ≥ (7/22) Σ_{k/2<p<11k/20} log p.

  • h_cheb (used, not proved, by GR: the "≫ k" on p. 25): Σ_{k/2<p<11k/20} log p ≥ k/100 for large k. This is the prime number theorem for the interval (x, 1.1x): θ(1.1x) − θ(x) ~ 0.1x with x = k/2, so the sum is ~ k/20. Bertrand's postulate is too weak and Chebyshev's elementary constants do not reach a ratio of 1.1.
  • h_sander ((5.1), p. 18, the J = 1 case of Lemma 5.1, which GR take from Sander [Sa2], J. reine angew. Math. 430): for every C and ε > 0, for large k on the range above, |Σ_{k/2<p<11k/20} ({n/p} − 1/2) log p| ≤ εk. From (5.1) with x = n and y = k/2, 11k/20: the error terms are y^{3/2+δ}x^{−1/2}log⁵x ≪ k^{1/2+2δ} (as n ≫ k²/log k) and y^{1−c(log y/log x)²}log⁵x ≪ k^{1−c/4+o(1)} (as n < k²), both o(k). GR's ψ is 0 at integers and {t} − 1/2 elsewhere; the two differ only at primes p | n, of which there are at most two in (k/2, k] when n < k² and k > 8, so the difference is O(log k). That is our second deviation in form.

Assembly (proved in Lean). With ε = 1/1000: (7/22)(k/100) > k/1000, a contradiction. So the assembly is elementary; both holes are analytic number theory (h_cheb: PNT in short intervals; h_sander: exponential sums over primes, Vaughan's identity).

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).

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