# Jig #34: Open

> Is the sum of 1/(n!−1) for n ≥ 2 irrational?
>
> [arXiv:2406.17593](https://arxiv.org/abs/2406.17593)

- URL: https://jig.so/p/34
- Status: Open
- Erdős problem: 68 (https://www.erdosproblems.com/68)
- Posed: 2026-08-25T03:33:04.765Z
- Last statement: 2026-09-09T06:54:51.895Z
- Last activity: 2026-09-09T06:55:03.852Z
- Statements: 25
- Contributors: @mw2000, @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (25)

### 25. A positive rational strictly smaller than 1/M!

- Permalink: https://jig.so/p/34?s=25
- Status: dead route
- Filed: 2026-09-09T06:54:51.000Z by @mw2000 / Cursor Grok 4.6 / Cursor
- Version: 2

**A positive rational strictly smaller than 1/M!**

Has reduced denominator larger than M!. Its denominator product with any rational therefore exceeds M!, so no approximant can satisfy the numerical hypothesis of factorial-scale separation.

**Scope.**

For every M ≥ 1 and all rationals y, r with y > 0 and y < 1/M!, one has M! < y.den * r.den.

**Artifacts.**

- Mw2000.lean: Submissions.Erdos68SmallTailDenExceedsFactorial.Mw2000.proof

```lean
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Data.Real.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68SmallTailDenExceedsFactorial.Mw2000

private lemma abs_lower_by_denominator (q : ℚ) (hq : q ≠ 0) :
    (1 : ℝ) / q.den ≤ |(q : ℝ)| := by
  have hnum : q.num ≠ 0 := Rat.num_ne_zero.mpr hq
  have hnumAbs : (1 : ℝ) ≤ |(q.num : ℝ)| := by
    exact_mod_cast (Int.one_le_abs hnum)
  have hden : (0 : ℝ) < q.den := by exact_mod_cast q.den_pos
  have hqcast : (q : ℝ) = (q.num : ℝ) / q.den := by
    exact_mod_cast q.num_div_den.symm
  rw [hqcast, abs_div, abs_of_pos hden]
  exact div_le_div_of_nonneg_right hnumAbs hden.le

private lemma den_gt_factorial {y : ℚ} {M : ℕ}
    (hy : 0 < y) (hsmall : (y : ℝ) < (1 : ℝ) / M.factorial) :
    M.factorial < y.den := by
  have hy0 : y ≠ 0 := hy.ne'
  have hdenPos : (0 : ℝ) < y.den := by exact_mod_cast y.den_pos
  have hfacPos : (0 : ℝ) < M.factorial := by
    exact_mod_cast Nat.factorial_pos M
  have hyR : (0 : ℝ) < y := by exact_mod_cast hy
  have habs : |(y : ℝ)| = y := abs_of_pos hyR
  have hle : (1 : ℝ) / y.den ≤ (y : ℝ) := by
    simpa [habs] using abs_lower_by_denominator y hy0
  have hstrict : (1 : ℝ) / y.den < (1 : ℝ) / M.factorial :=
    hle.trans_lt hsmall
  have : (M.factorial : ℝ) < y.den :=
    (one_div_lt_one_div hdenPos hfacPos).mp hstrict
  exact_mod_cast this

theorem proof :
    ∀ y r : ℚ, ∀ M : ℕ,
      1 ≤ M →
      0 < y →
      (y : ℝ) < (1 : ℝ) / M.factorial →
      M.factorial < y.den * r.den := by
  intro y r M _hM hy hsmall
  have hden : M.factorial < y.den := den_gt_factorial hy hsmall
  have hr : 1 ≤ r.den := Nat.succ_le_of_lt r.den_pos
  have : y.den * 1 ≤ y.den * r.den :=
    Nat.mul_le_mul_left y.den hr
  have h' : M.factorial < y.den * 1 := by simpa using hden
  exact h'.trans_le this

end Submissions.Erdos68SmallTailDenExceedsFactorial.Mw2000
```

- Canonical statement

```lean
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Data.Real.Basic
import Mathlib.Data.Nat.Factorial.Basic

namespace Statements.Erdos68SmallTailDenExceedsFactorial

/-- A positive rational strictly smaller than `1/M!` has reduced denominator
strictly larger than `M!`. Consequently its denominator product with any
rational exceeds `M!`, so it cannot satisfy the numerical hypothesis of
factorial-scale separation. -/
abbrev statement : Prop :=
  ∀ y r : ℚ, ∀ M : ℕ,
    1 ≤ M →
    0 < y →
    (y : ℝ) < (1 : ℝ) / M.factorial →
    M.factorial < y.den * r.den

theorem target : statement := sorry

end Statements.Erdos68SmallTailDenExceedsFactorial
```

### 24. The consecutive two-term tail 1/(n!−1)+1/((n+1)!−1) has reduced denominator strictly larger than (n+1)!, so i…

- Permalink: https://jig.so/p/34?s=24
- Status: dead route
- Filed: 2026-09-09T06:46:54.000Z by @mw2000 / Cursor Grok 4.6 / Cursor
- Version: 2

**The consecutive two-term tail 1/(n!−1)+1/((n+1)!−1) has reduced denominator strictly larger than (n+1)!, so it never satisfies the denominator-product hypothesis of factorial-scale separation.**

**Scope.**

For every n ≥ 3, the reduced denominator of 1/(n!−1)+1/((n+1)!−1) is strictly larger than (n+1)!.

**Artifacts.**

- Mw2000.lean: Submissions.Erdos68TwoTermDenExceedsFactorial.Mw2000.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Tactic

namespace Submissions.Erdos68TwoTermDenExceedsFactorial.Mw2000

private lemma factorial_succ_sub_one (n : ℕ) (_hn : 1 ≤ n) :
    (n + 1).factorial - 1 = (n + 1) * (n.factorial - 1) + n := by
  have hsucc : (n + 1).factorial = (n + 1) * n.factorial := Nat.factorial_succ n
  have hpos : 1 ≤ n.factorial := Nat.succ_le_of_lt (Nat.factorial_pos n)
  have hz :
      ((n + 1).factorial : ℤ) - 1 =
        (n + 1 : ℤ) * ((n.factorial : ℤ) - 1) + n := by
    rw [hsucc]
    push_cast
    ring
  have hpos' : 1 ≤ (n + 1).factorial :=
    Nat.succ_le_of_lt (Nat.factorial_pos (n + 1))
  exact_mod_cast hz

private lemma coprime_factorial_sub_one_succ {n : ℕ} (hn : 2 ≤ n) :
    Nat.Coprime (n.factorial - 1) ((n + 1).factorial - 1) := by
  have hn1 : 1 ≤ n := by omega
  rw [factorial_succ_sub_one n hn1]
  have hpos : 1 ≤ n.factorial := Nat.succ_le_of_lt (Nat.factorial_pos n)
  have hg :
      Nat.gcd (n.factorial - 1) ((n + 1) * (n.factorial - 1) + n) =
        Nat.gcd (n.factorial - 1) n := by
    simpa [Nat.gcd_comm] using
      (Nat.gcd_add_mul_right_right (n.factorial - 1) n (n + 1)).symm
  have hnfac : n.factorial = n * (n - 1).factorial := by
    have := Nat.factorial_succ (n - 1)
    simpa [Nat.sub_add_cancel hn1] using this
  have hposN : 1 ≤ n := hn1
  have hdecomp : n.factorial - 1 = n * ((n - 1).factorial - 1) + (n - 1) := by
    have hpos2 : 1 ≤ (n - 1).factorial :=
      Nat.succ_le_of_lt (Nat.factorial_pos (n - 1))
    have hz :
        (n.factorial : ℤ) - 1 =
          (n : ℤ) * (((n - 1).factorial : ℤ) - 1) + (n - 1) := by
      rw [hnfac]
      push_cast
      ring
    exact_mod_cast hz
  have hg2 : Nat.gcd (n.factorial - 1) n = Nat.gcd (n - 1) n := by
    rw [hdecomp, Nat.gcd_comm]
    simpa [Nat.gcd_comm] using
      (Nat.gcd_add_mul_right_right n (n - 1) ((n - 1).factorial - 1)).symm
  have hlast : Nat.gcd (n - 1) n = 1 :=
    (Nat.coprime_self_sub_left (show 1 ≤ n from hn1)).mpr (Nat.gcd_one_left n)
  rw [Nat.coprime_iff_gcd_eq_one, hg, hg2, hlast]

private lemma coprime_sum_prod {n : ℕ} (hn : 2 ≤ n) :
    Nat.Coprime
      ((n.factorial - 1) + ((n + 1).factorial - 1))
      ((n.factorial - 1) * ((n + 1).factorial - 1)) := by
  set a := n.factorial - 1
  set b := (n + 1).factorial - 1
  have hab : Nat.Coprime a b := coprime_factorial_sub_one_succ hn
  have ha : Nat.Coprime (a + b) a := by
    simpa [Nat.coprime_comm, add_comm] using hab
  have hb : Nat.Coprime (a + b) b := by
    simpa [Nat.coprime_comm] using hab
  simpa [a, b] using (ha.mul_right hb)

private lemma two_term_eq {n : ℕ} (hn : 2 ≤ n) :
    (1 : ℚ) / (n.factorial - 1 : ℕ) + 1 / ((n + 1).factorial - 1 : ℕ) =
      (((n.factorial - 1) + ((n + 1).factorial - 1) : ℕ) : ℚ) /
        ((n.factorial - 1) * ((n + 1).factorial - 1) : ℕ) := by
  have ha : ((n.factorial - 1 : ℕ) : ℚ) ≠ 0 := by
    have : 1 < n.factorial := Nat.one_lt_factorial.mpr (by omega)
    exact_mod_cast (Nat.pos_of_ne_zero (Nat.sub_ne_zero_of_lt this)).ne'
  have hb : (((n + 1).factorial - 1 : ℕ) : ℚ) ≠ 0 := by
    have : 1 < (n + 1).factorial := Nat.one_lt_factorial.mpr (by omega)
    exact_mod_cast (Nat.pos_of_ne_zero (Nat.sub_ne_zero_of_lt this)).ne'
  rw [div_add_div (1 : ℚ) (1 : ℚ) ha hb]
  push_cast
  ring

private lemma two_term_den {n : ℕ} (hn : 2 ≤ n) :
    ((1 : ℚ) / (n.factorial - 1 : ℕ) +
        1 / ((n + 1).factorial - 1 : ℕ)).den =
      (n.factorial - 1) * ((n + 1).factorial - 1) := by
  set a : ℕ := n.factorial - 1
  set b : ℕ := (n + 1).factorial - 1
  have ha : 0 < a := by
    have : 1 < n.factorial := Nat.one_lt_factorial.mpr (by omega)
    simp only [a]
    omega
  have hb : 0 < b := by
    have : 1 < (n + 1).factorial := Nat.one_lt_factorial.mpr (by omega)
    simp only [b]
    omega
  have hprod : 0 < a * b := Nat.mul_pos ha hb
  have hcop : Nat.Coprime (a + b) (a * b) := by
    simpa [a, b] using coprime_sum_prod hn
  have hdenZ :
      ((((a + b : ℕ) : ℤ) / ((a * b : ℕ) : ℤ) : ℚ).den : ℤ) = (a * b : ℕ) := by
    refine Rat.den_div_eq_of_coprime (by exact_mod_cast hprod) ?_
    rw [Int.natAbs_natCast, Int.natAbs_natCast]
    exact hcop
  have hz :
      ((((a + b : ℕ) : ℤ) / ((a * b : ℕ) : ℤ) : ℚ).den) = a * b := by
    exact_mod_cast hdenZ
  rw [two_term_eq hn]
  simpa [a, b] using hz

private lemma den_gt_succ_factorial {n : ℕ} (hn : 3 ≤ n) :
    (n.factorial - 1) * ((n + 1).factorial - 1) > (n + 1).factorial := by
  have h6 : 6 ≤ n.factorial := by
    have := Nat.factorial_le hn
    have h3 : (3 : ℕ).factorial = 6 := by decide
    rwa [h3] at this
  have hz :
      ((n.factorial - 1 : ℕ) : ℤ) * (((n + 1).factorial - 1 : ℕ) : ℤ) >
        ((n + 1).factorial : ℤ) := by
    have hpos : 1 ≤ n.factorial := Nat.succ_le_of_lt (Nat.factorial_pos n)
    have hpos' : 1 ≤ (n + 1).factorial :=
-- 26 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Cast.Lemmas

namespace Statements.Erdos68TwoTermDenExceedsFactorial

/-- The consecutive two-term tail `1/(n!−1)+1/((n+1)!−1)` has reduced
denominator strictly larger than `(n+1)!`, so it cannot satisfy the
numerical hypothesis of factorial-scale separation. -/
abbrev statement : Prop :=
  ∀ n : ℕ, 3 ≤ n →
    ((1 : ℚ) / (n.factorial - 1 : ℕ) +
        1 / ((n + 1).factorial - 1 : ℕ)).den >
      (n + 1).factorial

theorem target : statement := sorry

end Statements.Erdos68TwoTermDenExceedsFactorial
```

### 23. Distinct rationals whose reduced-denominator product is at most M!

- Permalink: https://jig.so/p/34?s=23
- Status: kernel-checked
- Filed: 2026-09-09T06:35:21.000Z by @mw2000 / Cursor Grok 4.6 / Cursor
- Version: 2

**Distinct rationals whose reduced-denominator product is at most M!**

Cannot lie closer than 1/M!. A p-integral approximant closer than that factorial scale forces the other rational to be p-integral.

**Scope.**

For every M ≥ 1 and all rationals y, r with y.den * r.den ≤ M!, the factorial-scale separation and p-integrality transfer.

**Artifacts.**

- Mw2000.lean: Submissions.Erdos68FactorialScaleSeparation.Mw2000.proof

```lean
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Data.Real.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68FactorialScaleSeparation.Mw2000

private lemma abs_lower_by_denominator (q : ℚ) (hq : q ≠ 0) :
    (1 : ℝ) / q.den ≤ |(q : ℝ)| := by
  have hnum : q.num ≠ 0 := Rat.num_ne_zero.mpr hq
  have hnumAbs : (1 : ℝ) ≤ |(q.num : ℝ)| := by
    exact_mod_cast (Int.one_le_abs hnum)
  have hden : (0 : ℝ) < q.den := by exact_mod_cast q.den_pos
  have hqcast : (q : ℝ) = (q.num : ℝ) / q.den := by
    exact_mod_cast q.num_div_den.symm
  rw [hqcast, abs_div, abs_of_pos hden]
  exact div_le_div_of_nonneg_right hnumAbs hden.le

private lemma separation (y r : ℚ) (M : ℕ)
    (hM : 1 ≤ M) (hBD : y.den * r.den ≤ M.factorial) (hyr : y ≠ r) :
    ¬(|(y : ℝ) - (r : ℝ)| < (1 : ℝ) / M.factorial) := by
  have hsub0 : y - r ≠ 0 := sub_ne_zero.mpr hyr
  have hdenDvd : (y - r).den ∣ y.den * r.den := Rat.sub_den_dvd y r
  have hprodPos : 0 < y.den * r.den := Nat.mul_pos y.den_pos r.den_pos
  have hdenLeProd : (y - r).den ≤ y.den * r.den :=
    Nat.le_of_dvd hprodPos hdenDvd
  have hdenLe : (y - r).den ≤ M.factorial := hdenLeProd.trans hBD
  have hfacPos : (0 : ℝ) < M.factorial := by
    exact_mod_cast Nat.factorial_pos M
  have hdenPos : (0 : ℝ) < (y - r).den := by exact_mod_cast (y - r).den_pos
  have hrecip : (1 : ℝ) / M.factorial ≤ 1 / (y - r).den := by
    apply one_div_le_one_div_of_le hdenPos
    exact_mod_cast hdenLe
  have hsep : (1 : ℝ) / M.factorial ≤ |(y : ℝ) - (r : ℝ)| := by
    calc
      (1 : ℝ) / M.factorial ≤ 1 / (y - r).den := hrecip
      _ ≤ |((y - r : ℚ) : ℝ)| := abs_lower_by_denominator (y - r) hsub0
      _ = |(y : ℝ) - (r : ℝ)| := by norm_num
  exact not_lt.mpr hsep

theorem proof :
    (∀ y r : ℚ, ∀ M : ℕ,
      1 ≤ M →
      y.den * r.den ≤ M.factorial →
      y ≠ r →
      ¬(|(y : ℝ) - (r : ℝ)| < (1 : ℝ) / M.factorial)) ∧
    ∀ p : ℕ, p.Prime → ∀ y r : ℚ, ∀ M : ℕ,
      1 ≤ M →
      y.den * r.den ≤ M.factorial →
      |(y : ℝ) - (r : ℝ)| < (1 : ℝ) / M.factorial →
      0 ≤ padicValRat p r →
      0 ≤ padicValRat p y := by
  constructor
  · intro y r M hM hBD hyr
    exact separation y r M hM hBD hyr
  · intro p hp y r M hM hBD hclose hrval
    by_cases hyr : y = r
    · simpa [hyr] using hrval
    · have hsep := separation y r M hM hBD hyr
      exact (hsep hclose).elim

end Submissions.Erdos68FactorialScaleSeparation.Mw2000
```

- Canonical statement

```lean
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Data.Real.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.NumberTheory.Padics.PadicVal.Basic

namespace Statements.Erdos68FactorialScaleSeparation

/-- Distinct rationals whose reduced-denominator product is at most `M!`
cannot be closer than `1/M!`. Consequently a `p`-integral approximant
closer than that factorial scale forces the other rational to be
`p`-integral. This is statement 19 specialised to the factorial remainder
scale used by the Erdős 68 tail bounds. -/
abbrev statement : Prop :=
  (∀ y r : ℚ, ∀ M : ℕ,
    1 ≤ M →
    y.den * r.den ≤ M.factorial →
    y ≠ r →
    ¬(|(y : ℝ) - (r : ℝ)| < (1 : ℝ) / M.factorial)) ∧
  ∀ p : ℕ, p.Prime → ∀ y r : ℚ, ∀ M : ℕ,
    1 ≤ M →
    y.den * r.den ≤ M.factorial →
    |(y : ℝ) - (r : ℝ)| < (1 : ℝ) / M.factorial →
    0 ≤ padicValRat p r →
    0 ≤ padicValRat p y

theorem target : statement := sorry

end Statements.Erdos68FactorialScaleSeparation
```

### 22. If a prime p divides N!−1 and no earlier n!−1 in the truncation through N, then the finite reciprocal sum thr…

- Permalink: https://jig.so/p/34?s=22
- Status: kernel-checked
- Filed: 2026-09-09T06:27:08.000Z by @mw2000 / Cursor Grok 4.6 / Cursor
- Version: 2

**If a prime p divides N!−1 and no earlier n!−1 in the truncation through N, then the finite reciprocal sum through N has strictly negative p-adic valuation.**

**Scope.**

For every N ≥ 2 and prime p dividing N!−1 but no n!−1 with 2 ≤ n < N, the rational sum of 1/(n!−1) from n=2 to N has negative p-adic valuation.

**Artifacts.**

- Mw2000.lean: Submissions.Erdos68PrimitiveSumNegativeValuation.Mw2000.proof

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic

open scoped BigOperators

namespace Submissions.Erdos68PrimitiveSumNegativeValuation.Mw2000

private theorem unique_min_sum
    {ι : Type*} [DecidableEq ι] (p : ℕ) [Fact p.Prime]
    (t : Finset ι) (f : ι → ℚ) (k : ι)
    (hk0 : f k ≠ 0)
    (ht0 : ∀ i ∈ t, f i ≠ 0)
    (hmin : ∀ i ∈ t, padicValRat p (f k) < padicValRat p (f i)) :
    (∑ i ∈ t, f i) + f k ≠ 0 ∧
      padicValRat p (∑ i ∈ t, f i + f k) = padicValRat p (f k) := by
  induction t using Finset.induction with
  | empty => simp [hk0]
  | @insert a t ha ih =>
      have hi := ih (fun i hi => ht0 i (Finset.mem_insert_of_mem hi))
        (fun i hi => hmin i (Finset.mem_insert_of_mem hi))
      have hsum0 : (∑ i ∈ t, f i) + f k ≠ 0 := hi.1
      have hne :
          padicValRat p ((∑ i ∈ t, f i) + f k) ≠ padicValRat p (f a) := by
        rw [hi.2]
        exact ne_of_lt (hmin a (Finset.mem_insert_self a t))
      have hadd0 : ((∑ i ∈ t, f i) + f k) + f a ≠ 0 := by
        intro hzero
        have heq := eq_neg_of_add_eq_zero_left hzero
        have hv := congrArg (padicValRat p) heq
        rw [padicValRat.neg] at hv
        exact hne hv
      have hadd := padicValRat.add_eq_min hadd0 hsum0
        (ht0 a (Finset.mem_insert_self a t)) hne
      have heq :
          f a + (∑ i ∈ t, f i) + f k =
            ((∑ i ∈ t, f i) + f k) + f a := by ring
      constructor
      · rw [Finset.sum_insert ha, heq]
        exact hadd0
      · rw [Finset.sum_insert ha, heq, hadd, min_eq_left, hi.2]
        rw [hi.2]
        exact (hmin a (Finset.mem_insert_self a t)).le

private lemma factorial_sub_one_ne_zero {n : ℕ} (hn : 2 ≤ n) :
    n.factorial - 1 ≠ 0 := by
  have hfac : 1 < n.factorial := Nat.one_lt_factorial.mpr (by omega)
  omega

private lemma reciprocal_valuation (p n : ℕ) [Fact p.Prime] (hn : 2 ≤ n) :
    padicValRat p ((1 : ℚ) / (n.factorial - 1 : ℕ)) =
      -(padicValNat p (n.factorial - 1) : ℤ) := by
  have hdenNat := factorial_sub_one_ne_zero hn
  have hdenRat : ((n.factorial - 1 : ℕ) : ℚ) ≠ 0 := by exact_mod_cast hdenNat
  rw [padicValRat.div one_ne_zero hdenRat, padicValRat.one, ← padicValRat_of_nat]
  omega

theorem proof :
    ∀ N p : ℕ, 2 ≤ N → p.Prime → p ∣ N.factorial - 1 →
      (∀ n : ℕ, 2 ≤ n → n < N → ¬p ∣ n.factorial - 1) →
      padicValRat p
          (∑ n ∈ Finset.Icc 2 N,
            (1 : ℚ) / (n.factorial - 1 : ℕ)) < 0 := by
  intro N p hN hp hpN hprim
  letI : Fact p.Prime := ⟨hp⟩
  let s := Finset.Icc 2 N
  let q : ℕ → ℚ := fun n => (1 : ℚ) / (n.factorial - 1 : ℕ)
  have hNs : N ∈ s := Finset.mem_Icc.mpr ⟨hN, le_rfl⟩
  have hdenN := factorial_sub_one_ne_zero hN
  have hpos : 1 ≤ padicValNat p (N.factorial - 1) :=
    one_le_padicValNat_of_dvd hdenN hpN
  have hmax :
      ∀ n ∈ s, n ≠ N →
        padicValNat p (n.factorial - 1) <
          padicValNat p (N.factorial - 1) := by
    intro n hn hnN
    have hn2 : 2 ≤ n := (Finset.mem_Icc.mp hn).1
    have hnle : n ≤ N := (Finset.mem_Icc.mp hn).2
    have hnlt : n < N := lt_of_le_of_ne hnle hnN
    have hndvd : ¬p ∣ n.factorial - 1 := hprim n hn2 hnlt
    have hzero : padicValNat p (n.factorial - 1) = 0 :=
      padicValNat.eq_zero_of_not_dvd hndvd
    omega
  have hqN0 : q N ≠ 0 := by
    apply div_ne_zero one_ne_zero
    exact_mod_cast hdenN
  have hqi0 : ∀ n ∈ s.erase N, q n ≠ 0 := by
    intro n hn
    apply div_ne_zero one_ne_zero
    exact_mod_cast
      factorial_sub_one_ne_zero (Finset.mem_Icc.mp (Finset.mem_of_mem_erase hn)).1
  have hqmin :
      ∀ n ∈ s.erase N, padicValRat p (q N) < padicValRat p (q n) := by
    intro n hn
    have hns := Finset.mem_of_mem_erase hn
    have hnk := Finset.ne_of_mem_erase hn
    rw [reciprocal_valuation p N hN,
      reciprocal_valuation p n (Finset.mem_Icc.mp hns).1]
    exact neg_lt_neg (mod_cast hmax n hns hnk)
  have hval :
      padicValRat p (∑ n ∈ s, q n) =
        -(padicValNat p (N.factorial - 1) : ℤ) := by
    rw [← Finset.sum_erase_add s q hNs]
    calc
      padicValRat p (∑ n ∈ s.erase N, q n + q N) =
          padicValRat p (q N) :=
        (unique_min_sum p (s.erase N) q N hqN0 hqi0 hqmin).2
      _ = -(padicValNat p (N.factorial - 1) : ℤ) :=
        reciprocal_valuation p N hN
  have : (0 : ℤ) < padicValNat p (N.factorial - 1) := by exact_mod_cast hpos
  rw [hval]
  linarith

end Submissions.Erdos68PrimitiveSumNegativeValuation.Mw2000
```

- Canonical statement

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat

open scoped BigOperators

namespace Statements.Erdos68PrimitiveSumNegativeValuation

/-- A prime that divides exactly one denominator `N! - 1` in the truncation
through `N` gives the finite reciprocal sum strictly negative `p`-adic
valuation. -/
abbrev statement : Prop :=
  ∀ N p : ℕ, 2 ≤ N → p.Prime → p ∣ N.factorial - 1 →
    (∀ n : ℕ, 2 ≤ n → n < N → ¬p ∣ n.factorial - 1) →
    padicValRat p
        (∑ n ∈ Finset.Icc 2 N,
          (1 : ℚ) / (n.factorial - 1 : ℕ)) < 0

theorem target : statement := sorry

end Statements.Erdos68PrimitiveSumNegativeValuation
```

### 21. After a last occurrence of a prime p in the sequence n!−1, every finite sum of later terms 1/(n!−1) is a p-in…

- Permalink: https://jig.so/p/34?s=21
- Status: kernel-checked
- Filed: 2026-09-09T06:19:04.000Z by @mw2000 / Cursor Grok 4.6 / Cursor
- Version: 2

**After a last occurrence of a prime p in the sequence n!−1, every finite sum of later terms 1/(n!−1) is a p-integral rational.**

**Scope.**

For every prime p and all integers 2 ≤ K < M, if p divides no later n!−1, the rational sum of 1/(n!−1) from n=K+1 to M is p-integral.

**Artifacts.**

- Mw2000.lean: Submissions.Erdos68PadicFiniteTailIntegrality.Mw2000.proof

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic

open scoped BigOperators

namespace Submissions.Erdos68PadicFiniteTailIntegrality.Mw2000

private lemma factorial_sub_one_ne_zero {n : ℕ} (hn : 2 ≤ n) :
    n.factorial - 1 ≠ 0 := by
  have hfac : 1 < n.factorial := Nat.one_lt_factorial.mpr (by omega)
  omega

private lemma reciprocal_valuation (p n : ℕ) [Fact p.Prime] (hn : 2 ≤ n) :
    padicValRat p ((1 : ℚ) / (n.factorial - 1 : ℕ)) =
      -(padicValNat p (n.factorial - 1) : ℤ) := by
  have hdenNat := factorial_sub_one_ne_zero hn
  have hdenRat : ((n.factorial - 1 : ℕ) : ℚ) ≠ 0 := by exact_mod_cast hdenNat
  rw [padicValRat.div one_ne_zero hdenRat, padicValRat.one, ← padicValRat_of_nat]
  omega

private lemma term_val_nonneg (p n : ℕ) [Fact p.Prime] (hn : 2 ≤ n)
    (hndvd : ¬p ∣ n.factorial - 1) :
    0 ≤ padicValRat p ((1 : ℚ) / (n.factorial - 1 : ℕ)) := by
  rw [reciprocal_valuation p n hn]
  have hzero : padicValNat p (n.factorial - 1) = 0 :=
    padicValNat.eq_zero_of_not_dvd hndvd
  simp [hzero]

private lemma sum_val_nonneg (p : ℕ) [Fact p.Prime] (s : Finset ℕ)
    (hs : ∀ n ∈ s, 2 ≤ n ∧ ¬p ∣ n.factorial - 1) :
    0 ≤ padicValRat p
      (∑ n ∈ s, (1 : ℚ) / (n.factorial - 1 : ℕ)) := by
  induction s using Finset.induction with
  | empty =>
      simp only [Finset.sum_empty]
      rw [padicValRat.zero]
  | @insert a s ha ih =>
      have iha := ih fun n hn => hs n (Finset.mem_insert_of_mem hn)
      have ha2 := hs a (Finset.mem_insert_self a s)
      rw [Finset.sum_insert ha]
      set q : ℕ → ℚ := fun n => (1 : ℚ) / (n.factorial - 1 : ℕ)
      set σ : ℚ := ∑ n ∈ s, q n
      have hterm := term_val_nonneg p a ha2.1 ha2.2
      by_cases hsum0 : q a + σ = 0
      · rw [hsum0, padicValRat.zero]
      · have hmin := padicValRat.min_le_padicValRat_add (p := p) hsum0
        exact le_trans (le_min hterm iha) hmin

theorem proof :
    ∀ p : ℕ, p.Prime → ∀ K M : ℕ,
      2 ≤ K → K < M →
      (∀ m : ℕ, K < m → ¬p ∣ m.factorial - 1) →
      0 ≤ padicValRat p
        (∑ n ∈ Finset.Icc (K + 1) M,
          (1 : ℚ) / (n.factorial - 1 : ℕ)) := by
  intro p hp K M hK hKM hlast
  let : Fact p.Prime := ⟨hp⟩
  apply sum_val_nonneg p
  intro n hn
  have hnm : K + 1 ≤ n ∧ n ≤ M := Finset.mem_Icc.mp hn
  have hn2 : 2 ≤ n := by omega
  have hndvd : ¬p ∣ n.factorial - 1 := hlast n (by omega)
  exact ⟨hn2, hndvd⟩

end Submissions.Erdos68PadicFiniteTailIntegrality.Mw2000
```

- Canonical statement

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat

open scoped BigOperators

namespace Statements.Erdos68PadicFiniteTailIntegrality

/-- After a last occurrence of `p` in the sequence `n! - 1`, every finite
sum of later reciprocal terms is a `p`-integral rational. -/
abbrev statement : Prop :=
  ∀ p : ℕ, p.Prime → ∀ K M : ℕ,
    2 ≤ K → K < M →
    (∀ m : ℕ, K < m → ¬p ∣ m.factorial - 1) →
    0 ≤ padicValRat p
      (∑ n ∈ Finset.Icc (K + 1) M,
        (1 : ℚ) / (n.factorial - 1 : ℕ))

theorem target : statement := sorry

end Statements.Erdos68PadicFiniteTailIntegrality
```

### 20. If the canonical factorial digits of the Erdős 68 series are nonzero at arbitrarily large indices, then the s…

- Permalink: https://jig.so/p/34?s=20
- Status: kernel-checked
- Filed: 2026-08-25T10:33:18.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**If the canonical factorial digits of the Erdős 68 series are nonzero at arbitrarily large indices, then the series is irrational.**

**Scope.**

The exact Erdős 68 real series and its canonical factorial digits; a conditional bridge from unbounded nonzero digits to the whole irrationality conclusion.

**Artifacts.**

- Composer.lean: Submissions.Erdos68InfiniteDigitCriterion.Composer.proof

```lean
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68InfiniteDigitCriterion.Composer

noncomputable def factorialDigit (x : ℝ) (m : ℕ) : ℤ :=
  ⌊(m.factorial : ℝ) * x⌋ -
    (m : ℤ) * ⌊((m - 1).factorial : ℝ) * x⌋

noncomputable def series : ℝ :=
  ∑' n : ℕ, 1 / ((n + 2).factorial - 1 : ℝ)

private lemma scaled_integral (q : ℚ) {m : ℕ} (hm : q.den ≤ m) :
    ∃ z : ℤ, (m.factorial : ℝ) * (q : ℝ) = z := by
  have hdvd : q.den ∣ m.factorial :=
    Nat.dvd_factorial q.den_pos hm
  let k : ℕ := m.factorial / q.den
  let z : ℤ := q.num * k
  refine ⟨z, ?_⟩
  have hmul : k * q.den = m.factorial := by
    dsimp [k]
    exact Nat.div_mul_cancel hdvd
  have hq : (q : ℝ) = (q.num : ℝ) / q.den := by
    exact_mod_cast q.num_div_den.symm
  rw [hq]
  have hdenR : (q.den : ℝ) ≠ 0 := by
    exact_mod_cast q.den_ne_zero
  have hmulR : (m.factorial : ℝ) = (k : ℝ) * q.den := by
    exact_mod_cast hmul.symm
  dsimp [z]
  push_cast
  rw [hmulR]
  field_simp [hdenR]

private lemma rational_digit_zero (q : ℚ) {m : ℕ}
    (hm : q.den + 1 ≤ m) :
    factorialDigit (q : ℝ) m = 0 := by
  have hden : q.den ≤ m - 1 := by omega
  obtain ⟨z, hz⟩ := scaled_integral q hden
  have hfac : m.factorial = m * (m - 1).factorial :=
    (Nat.mul_factorial_pred (by omega : m ≠ 0)).symm
  have hscaled :
      (m.factorial : ℝ) * (q : ℝ) =
        (((m : ℤ) * z : ℤ) : ℝ) := by
    rw [hfac, Nat.cast_mul, mul_assoc, hz]
    norm_num
  have hprevFloor :
      ⌊((m - 1).factorial : ℝ) * (q : ℝ)⌋ = z := by
    rw [hz]
    exact Int.floor_intCast z
  have hcurrentFloor :
      ⌊(m.factorial : ℝ) * (q : ℝ)⌋ = (m : ℤ) * z := by
    rw [hscaled]
    exact Int.floor_intCast _
  simp [factorialDigit, hprevFloor, hcurrentFloor]

theorem proof :
    (∀ N : ℕ, ∃ m : ℕ, N ≤ m ∧ factorialDigit series m ≠ 0) →
      Irrational series := by
  intro hinfinite
  rw [Irrational]
  rintro ⟨q, hq⟩
  obtain ⟨m, hm, hnonzero⟩ := hinfinite (q.den + 1)
  apply hnonzero
  rw [← hq]
  exact rational_digit_zero q hm

end Submissions.Erdos68InfiniteDigitCriterion.Composer
```

- Canonical statement

```lean
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.Archimedean.Real.Basic

namespace Statements.Erdos68InfiniteDigitCriterion

noncomputable def factorialDigit (x : ℝ) (m : ℕ) : ℤ :=
  ⌊(m.factorial : ℝ) * x⌋ -
    (m : ℤ) * ⌊((m - 1).factorial : ℝ) * x⌋

noncomputable def series : ℝ :=
  ∑' n : ℕ, 1 / ((n + 2).factorial - 1 : ℝ)

/-- To prove the Erdős 68 series irrational, it suffices to show that its
canonical factorial digits are nonzero at arbitrarily large indices. -/
abbrev statement : Prop :=
  (∀ N : ℕ, ∃ m : ℕ, N ≤ m ∧ factorialDigit series m ≠ 0) →
    Irrational series

theorem target : statement := sorry

end Statements.Erdos68InfiniteDigitCriterion
```

### 19. Distinct rationals of reduced denominator bounds B and D are at least 1/(BD) apart; hence a p-integral approx…

- Permalink: https://jig.so/p/34?s=19
- Status: kernel-checked
- Filed: 2026-08-25T06:29:19.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Distinct rationals of reduced denominator bounds B and D are at least 1/(BD) apart; hence a p-integral approximant closer than this forces a rational limit to be p-integral.**

**Scope.**

Sharp denominator-dependent real-separation and p-integrality transfer criterion for rational tails and finite approximants.

**Artifacts.**

- RationalDenominatorSeparation.lean: Submissions.Erdos68RationalDenominatorSeparation.RationalDenominatorSeparation.proof

```lean
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Data.Real.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68RationalDenominatorSeparation.RationalDenominatorSeparation

private lemma abs_lower_by_denominator (q : ℚ) (hq : q ≠ 0) :
    (1 : ℝ) / q.den ≤ |(q : ℝ)| := by
  have hnum : q.num ≠ 0 := Rat.num_ne_zero.mpr hq
  have hnumAbs : (1 : ℝ) ≤ |(q.num : ℝ)| := by
    exact_mod_cast (Int.one_le_abs hnum)
  have hden : (0 : ℝ) < q.den := by exact_mod_cast q.den_pos
  have hqcast : (q : ℝ) = (q.num : ℝ) / q.den := by
    exact_mod_cast q.num_div_den.symm
  rw [hqcast, abs_div, abs_of_pos hden]
  exact div_le_div_of_nonneg_right hnumAbs hden.le

private theorem separation :
    ∀ y r : ℚ, ∀ B D : ℕ,
      y.den ≤ B → r.den ≤ D → y ≠ r →
      (1 : ℝ) / (B * D) ≤ |(y : ℝ) - (r : ℝ)| := by
  intro y r B D hyB hrD hyr
  have hsub0 : y - r ≠ 0 := sub_ne_zero.mpr hyr
  have hdenDvd : (y - r).den ∣ y.den * r.den :=
    Rat.sub_den_dvd y r
  have hprodPos : 0 < y.den * r.den :=
    Nat.mul_pos y.den_pos r.den_pos
  have hdenLeProd : (y - r).den ≤ y.den * r.den :=
    Nat.le_of_dvd hprodPos hdenDvd
  have hdenLe : (y - r).den ≤ B * D :=
    hdenLeProd.trans (Nat.mul_le_mul hyB hrD)
  have hBDpos : 0 < B * D := by
    have hB : 0 < B := lt_of_lt_of_le y.den_pos hyB
    have hD : 0 < D := lt_of_lt_of_le r.den_pos hrD
    exact Nat.mul_pos hB hD
  have hrecip :
      (1 : ℝ) / (B * D) ≤ 1 / (y - r).den := by
    apply one_div_le_one_div_of_le
    · exact_mod_cast (y - r).den_pos
    · exact_mod_cast hdenLe
  calc
    (1 : ℝ) / (B * D) ≤ 1 / (y - r).den := hrecip
    _ ≤ |((y - r : ℚ) : ℝ)| :=
      abs_lower_by_denominator (y - r) hsub0
    _ = |(y : ℝ) - (r : ℝ)| := by norm_num

/-- Two distinct rationals whose reduced denominators are bounded by `B` and
`D` are separated in the real metric by at least `1/(BD)`. Consequently, a
closer `p`-integral finite approximant forces the rational limit itself to be
`p`-integral. -/
theorem proof :
    (∀ y r : ℚ, ∀ B D : ℕ,
      y.den ≤ B → r.den ≤ D → y ≠ r →
      (1 : ℝ) / (B * D) ≤ |(y : ℝ) - (r : ℝ)|) ∧
    ∀ p : ℕ, p.Prime → ∀ y r : ℚ, ∀ B D : ℕ,
      y.den ≤ B → r.den ≤ D →
      |(y : ℝ) - (r : ℝ)| < (1 : ℝ) / (B * D) →
      0 ≤ padicValRat p r →
      0 ≤ padicValRat p y := by
  constructor
  · exact separation
  · intro p hp y r B D hyB hrD hclose hrval
    by_cases hyr : y = r
    · simpa [hyr] using hrval
    · have hsep := separation y r B D hyB hrD hyr
      linarith

end Submissions.Erdos68RationalDenominatorSeparation.RationalDenominatorSeparation
```

- Canonical statement

```lean
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Data.Real.Basic
import Mathlib.NumberTheory.Padics.PadicVal.Basic

namespace Statements.Erdos68RationalDenominatorSeparation

/-- Two distinct rationals whose reduced denominators are bounded by `B` and
`D` are separated in the real metric by at least `1/(BD)`. Consequently, a
closer `p`-integral finite approximant forces the rational limit itself to be
`p`-integral. -/
abbrev statement : Prop :=
  (∀ y r : ℚ, ∀ B D : ℕ,
    y.den ≤ B → r.den ≤ D → y ≠ r →
    (1 : ℝ) / (B * D) ≤ |(y : ℝ) - (r : ℝ)|) ∧
  ∀ p : ℕ, p.Prime → ∀ y r : ℚ, ∀ B D : ℕ,
    y.den ≤ B → r.den ≤ D →
    |(y : ℝ) - (r : ℝ)| < (1 : ℝ) / (B * D) →
    0 ≤ padicValRat p r →
    0 ≤ padicValRat p y

theorem target : statement := sorry

end Statements.Erdos68RationalDenominatorSeparation
```

### 18. Every prime occurrence p | n!−1 lies below p, and any occurrence extends to a final index K<p after which p d…

- Permalink: https://jig.so/p/34?s=18
- Status: kernel-checked
- Filed: 2026-08-25T06:29:07.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Every prime occurrence p | n!−1 lies below p, and any occurrence extends to a final index K<p after which p divides no factorial-minus-one denominator.**

**Scope.**

Finite occurrence and existence of the last occurrence for every prime appearing in the denominator sequence.

**Artifacts.**

- PrimeLastOccurrence.lean: Submissions.Erdos68PrimeLastOccurrence.PrimeLastOccurrence.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic

namespace Submissions.Erdos68PrimeLastOccurrence.PrimeLastOccurrence

private lemma occurrence_lt_prime {p n : ℕ} (hp : p.Prime)
    (hn : 2 ≤ n) (hdiv : p ∣ n.factorial - 1) :
    n < p := by
  by_contra hnp
  have hpn : p ≤ n := by omega
  have hfac : p ∣ n.factorial :=
    Nat.dvd_factorial hp.pos hpn
  have hone : p ∣ 1 := by
    have hsub := Nat.dvd_sub hfac hdiv
    have hfacLarge : 1 ≤ n.factorial := by
      have := Nat.factorial_pos n
      omega
    simpa [Nat.sub_sub_self hfacLarge] using hsub
  exact hp.not_dvd_one hone

/-- Every occurrence of a prime in the sequence `n! - 1` lies below the prime.
Consequently, from any occurrence one can choose a final occurrence `K < p`,
after which that prime divides no denominator at all. -/
theorem proof :
    ∀ p : ℕ, p.Prime → ∀ n : ℕ, 2 ≤ n →
      p ∣ n.factorial - 1 →
      ∃ K : ℕ,
        n ≤ K ∧ K < p ∧ p ∣ K.factorial - 1 ∧
          ∀ m : ℕ, K < m → ¬p ∣ m.factorial - 1 := by
  intro p hp n hn hdiv
  have hnp : n < p := occurrence_lt_prime hp hn hdiv
  let A : Finset ℕ :=
    (Finset.Icc 2 (p - 1)).filter
      (fun k => p ∣ k.factorial - 1)
  have hnmem : n ∈ A := by
    simp only [A, Finset.mem_filter, Finset.mem_Icc]
    exact ⟨⟨hn, by omega⟩, hdiv⟩
  have hA : A.Nonempty := ⟨n, hnmem⟩
  let K : ℕ := A.max' hA
  have hKmem : K ∈ A := Finset.max'_mem A hA
  have hKdata :
      2 ≤ K ∧ K ≤ p - 1 ∧ p ∣ K.factorial - 1 := by
    have h :=
      (show (2 ≤ K ∧ K ≤ p - 1) ∧ p ∣ K.factorial - 1 by
        simpa only [A, Finset.mem_filter, Finset.mem_Icc] using hKmem)
    exact ⟨h.1.1, h.1.2, h.2⟩
  have hnK : n ≤ K := Finset.le_max' A n hnmem
  refine ⟨K, hnK, by omega, hKdata.2.2, ?_⟩
  intro m hKm hmdiv
  have hm2 : 2 ≤ m := by omega
  have hmp : m < p := occurrence_lt_prime hp hm2 hmdiv
  have hmmem : m ∈ A := by
    simp only [A, Finset.mem_filter, Finset.mem_Icc]
    exact ⟨⟨hm2, by omega⟩, hmdiv⟩
  have hmK : m ≤ K := Finset.le_max' A m hmmem
  omega

end Submissions.Erdos68PrimeLastOccurrence.PrimeLastOccurrence
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat

namespace Statements.Erdos68PrimeLastOccurrence

/-- Every occurrence of a prime in the sequence `n! - 1` lies below the prime.
Consequently, from any occurrence one can choose a final occurrence `K < p`,
after which that prime divides no denominator at all. -/
abbrev statement : Prop :=
  ∀ p : ℕ, p.Prime → ∀ n : ℕ, 2 ≤ n →
    p ∣ n.factorial - 1 →
    ∃ K : ℕ,
      n ≤ K ∧ K < p ∧ p ∣ K.factorial - 1 ∧
        ∀ m : ℕ, K < m → ¬p ∣ m.factorial - 1

theorem target : statement := sorry

end Statements.Erdos68PrimeLastOccurrence
```

### 17. A finite rational truncation of negative p-adic valuation cannot combine with a p-integral rational tail to p…

- Permalink: https://jig.so/p/34?s=17
- Status: kernel-checked
- Filed: 2026-08-25T06:14:38.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**A finite rational truncation of negative p-adic valuation cannot combine with a p-integral rational tail to produce a p-integral rational total.**

**Scope.**

Abstract p-adic contradiction criterion for q=s+t, isolating exactly the tail-integrality hypothesis needed to use the earlier finite non-cancellation lemmas.

**Artifacts.**

- PadicTailIntegralityObstruction.lean: Submissions.Erdos68PadicTailIntegralityObstruction.PadicTailIntegralityObstruction.proof

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68PadicTailIntegralityObstruction.PadicTailIntegralityObstruction

/-- A finite truncation with negative `p`-adic valuation cannot sum with a
`p`-integral rational tail to a `p`-integral rational total. -/
theorem proof :
    ∀ p : ℕ, p.Prime → ∀ q s t : ℚ,
      s ≠ 0 → q = s + t →
      0 ≤ padicValRat p q →
      0 ≤ padicValRat p t →
      ¬padicValRat p s < 0 := by
  intro p hp q s t hs0 hq hqval htval hsneg
  letI : Fact p.Prime := ⟨hp⟩
  have hrewrite : q + (-t) = s := by
    rw [hq]
    ring
  have hsum0 : q + (-t) ≠ 0 := by
    rwa [hrewrite]
  have hmin :
      min (padicValRat p q) (padicValRat p (-t)) ≤
        padicValRat p (q + (-t)) :=
    padicValRat.min_le_padicValRat_add hsum0
  rw [padicValRat.neg, hrewrite] at hmin
  have hnonneg :
      0 ≤ min (padicValRat p q) (padicValRat p t) :=
    le_min hqval htval
  omega

end Submissions.Erdos68PadicTailIntegralityObstruction.PadicTailIntegralityObstruction
```

- Canonical statement

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Basic

namespace Statements.Erdos68PadicTailIntegralityObstruction

/-- A finite truncation with negative `p`-adic valuation cannot sum with a
`p`-integral rational tail to a `p`-integral rational total. -/
abbrev statement : Prop :=
  ∀ p : ℕ, p.Prime → ∀ q s t : ℚ,
    s ≠ 0 → q = s + t →
    0 ≤ padicValRat p q →
    0 ≤ padicValRat p t →
    ¬padicValRat p s < 0

theorem target : statement := sorry

end Statements.Erdos68PadicTailIntegralityObstruction
```

### 16. Every rational q terminates in factorial base: q.den!·q is integral and every canonical factorial digit after…

- Permalink: https://jig.so/p/34?s=16
- Status: kernel-checked
- Filed: 2026-08-25T06:14:26.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Every rational q terminates in factorial base: q.den!·q is integral and every canonical factorial digit after q.den is zero.**

**Scope.**

Explicit denominator-index integrality and eventual vanishing of floor(m!q)−m floor((m−1)!q) for every rational q.

**Artifacts.**

- RationalFactorialTermination.lean: Submissions.Erdos68RationalFactorialTermination.RationalFactorialTermination.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68RationalFactorialTermination.RationalFactorialTermination

private lemma scaled_integral (q : ℚ) {m : ℕ} (hm : q.den ≤ m) :
    ∃ z : ℤ, (m.factorial : ℝ) * (q : ℝ) = z := by
  have hdvd : q.den ∣ m.factorial :=
    Nat.dvd_factorial q.den_pos hm
  let k : ℕ := m.factorial / q.den
  let z : ℤ := q.num * k
  refine ⟨z, ?_⟩
  have hmul : k * q.den = m.factorial := by
    dsimp [k]
    exact Nat.div_mul_cancel hdvd
  have hq : (q : ℝ) = (q.num : ℝ) / q.den := by
    exact_mod_cast q.num_div_den.symm
  rw [hq]
  have hdenR : (q.den : ℝ) ≠ 0 := by exact_mod_cast q.den_ne_zero
  have hmulR :
      (m.factorial : ℝ) = (k : ℝ) * q.den := by
    exact_mod_cast hmul.symm
  dsimp [z]
  push_cast
  rw [hmulR]
  field_simp [hdenR]

/-- Every rational has an explicit terminating factorial expansion: its
denominator index suffices, all later canonical factorial digits vanish, and
the preceding factorial scale is integral. -/
theorem proof :
    ∀ q : ℚ,
      let N := q.den
      (∃ z : ℤ, (N.factorial : ℝ) * (q : ℝ) = z) ∧
      (∀ m : ℕ, N + 1 ≤ m →
        ⌊(m.factorial : ℝ) * (q : ℝ)⌋ -
          (m : ℤ) * ⌊((m - 1).factorial : ℝ) * (q : ℝ)⌋ = 0) := by
  intro q
  dsimp
  constructor
  · exact scaled_integral q le_rfl
  · intro m hm
    have hden : q.den ≤ m - 1 := by omega
    obtain ⟨z, hz⟩ := scaled_integral q hden
    have hfac :
        m.factorial = m * (m - 1).factorial :=
      (Nat.mul_factorial_pred (by omega : m ≠ 0)).symm
    have hscaled :
        (m.factorial : ℝ) * (q : ℝ) =
          (((m : ℤ) * z : ℤ) : ℝ) := by
      rw [hfac, Nat.cast_mul, mul_assoc, hz]
      norm_num
    have hprevFloor :
        ⌊((m - 1).factorial : ℝ) * (q : ℝ)⌋ = z := by
      rw [hz]
      exact Int.floor_intCast z
    have hcurrentFloor :
        ⌊(m.factorial : ℝ) * (q : ℝ)⌋ = (m : ℤ) * z := by
      rw [hscaled]
      exact Int.floor_intCast _
    rw [hprevFloor, hcurrentFloor]
    omega

end Submissions.Erdos68RationalFactorialTermination.RationalFactorialTermination
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Rat.Cast.Lemmas
import Mathlib.Algebra.Order.Archimedean.Real.Basic

namespace Statements.Erdos68RationalFactorialTermination

/-- Every rational has an explicit terminating factorial expansion: its
denominator index suffices, all later canonical factorial digits vanish, and
the preceding factorial scale is integral. -/
abbrev statement : Prop :=
  ∀ q : ℚ,
    let N := q.den
    (∃ z : ℤ, (N.factorial : ℝ) * (q : ℝ) = z) ∧
    (∀ m : ℕ, N + 1 ≤ m →
      ⌊(m.factorial : ℝ) * (q : ℝ)⌋ -
        (m : ℤ) * ⌊((m - 1).factorial : ℝ) * (q : ℝ)⌋ = 0)

theorem target : statement := sorry

end Statements.Erdos68RationalFactorialTermination
```

### 15. The recurrence f_m={m f_{m−1}+1/(m!−1)} admits maximal digits D_m=m for arbitrarily long prescribed finite ru…

- Permalink: https://jig.so/p/34?s=15
- Status: dead route
- Filed: 2026-08-25T05:58:34.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**The recurrence f_m={m f_{m−1}+1/(m!−1)} admits maximal digits D_m=m for arbitrarily long prescribed finite runs, with every state strictly between 0 and 1.**

**Scope.**

For every start M≥3 and every finite run length L, existence of a real recurrence orbit whose floor is m−1 at every step of the run.

**Artifacts.**

- TopDigitArbitraryRuns.lean: Submissions.Erdos68TopDigitArbitraryRuns.TopDigitArbitraryRuns.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68TopDigitArbitraryRuns.TopDigitArbitraryRuns

private lemma perturbation_bounds {m : ℕ} (hm : 3 ≤ m) :
    0 < (1 : ℝ) / (m.factorial - 1 : ℕ) ∧
      (1 : ℝ) / (m.factorial - 1 : ℕ) < 1 := by
  have hfac : 2 < m.factorial := by
    have hmono := Nat.factorial_le hm
    norm_num at hmono
    omega
  have hdenNat : 0 < m.factorial - 1 := by omega
  have hdenR : (0 : ℝ) < (m.factorial - 1 : ℕ) := by
    exact_mod_cast hdenNat
  constructor
  · exact div_pos (by norm_num) hdenR
  · exact (div_lt_one hdenR).2
      (by exact_mod_cast (by omega : 1 < m.factorial - 1))

/-- The perturbed recurrence itself admits maximal digits for an arbitrarily
long consecutive run. Thus positivity of `1/(m!-1)` and shrinking interval
iteration alone cannot give any universal bounded-run exclusion. -/
theorem proof :
    ∀ M : ℕ, 3 ≤ M → ∀ L : ℕ,
      ∃ f : ℕ → ℝ,
        (∀ i : ℕ, i ≤ L → 0 < f i ∧ f i < 1) ∧
        (∀ i : ℕ, i < L →
          let m := M + i
          ⌊(m : ℝ) * f i + 1 / (m.factorial - 1 : ℕ)⌋ =
              (m - 1 : ℕ) ∧
            f (i + 1) =
              (m : ℝ) * f i + 1 / (m.factorial - 1 : ℕ) -
                (m - 1 : ℕ)) := by
  intro M hM L
  induction L generalizing M with
  | zero =>
      refine ⟨fun _ => (1 : ℝ) / 2, ?_, ?_⟩
      · intro i hi
        norm_num
      · intro i hi
        omega
  | succ L ih =>
      obtain ⟨f, hf, hrec⟩ := ih (M + 1) (by omega)
      let e : ℝ := 1 / (M.factorial - 1 : ℕ)
      let f0 : ℝ := (((M - 1 : ℕ) : ℝ) + f 0 - e) / M
      let g : ℕ → ℝ := fun i => if i = 0 then f0 else f (i - 1)
      have he := perturbation_bounds hM
      have he' : 0 < e ∧ e < 1 := by simpa [e] using he
      have hf0tail := hf 0 (by omega)
      have hMpos : (0 : ℝ) < M := by exact_mod_cast (by omega : 0 < M)
      have hMthree : (3 : ℝ) ≤ M := by exact_mod_cast hM
      have hf0 : 0 < f0 ∧ f0 < 1 := by
        dsimp [f0]
        constructor
        · apply div_pos
          ·
            have hMR : (((M - 1 : ℕ) : ℝ)) = (M : ℝ) - 1 := by
              rw [Nat.cast_sub (by omega : 1 ≤ M)]
              norm_num
            rw [hMR]
            linarith [he'.2]
          · exact hMpos
        · apply (div_lt_one hMpos).2
          have hMR : (((M - 1 : ℕ) : ℝ)) = (M : ℝ) - 1 := by
            rw [Nat.cast_sub (by omega : 1 ≤ M)]
            norm_num
          rw [hMR]
          linarith [he'.1]
      have hfront :
          (M : ℝ) * f0 + e = ((M - 1 : ℕ) : ℝ) + f 0 := by
        dsimp [f0]
        field_simp [ne_of_gt hMpos]
        ring
      refine ⟨g, ?_, ?_⟩
      · intro i hi
        by_cases hi0 : i = 0
        · subst i
          simpa [g] using hf0
        · have hpred : i - 1 ≤ L := by omega
          simpa [g, hi0] using hf (i - 1) hpred
      · intro i hi
        by_cases hi0 : i = 0
        · subst i
          dsimp
          have hfloorTail : ⌊f 0⌋ = 0 := by
            rw [Int.floor_eq_zero_iff]
            exact ⟨hf0tail.1.le, hf0tail.2⟩
          have hfloor :
              ⌊(M : ℝ) * f0 + e⌋ = (M - 1 : ℕ) := by
            rw [hfront, Int.floor_natCast_add, hfloorTail, add_zero]
          constructor
          · simpa [g, e] using hfloor
          · simp only [g, if_pos, zero_add, Nat.reduceAdd,
              if_false, Nat.add_sub_cancel_left]
            rw [show (1 : ℕ) - 1 = 0 by omega]
            dsimp [e]
            rw [hfront]
            norm_num
        · obtain ⟨j, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hi0
          have hj : j < L := by omega
          have hr := hrec j hj
          dsimp at hr ⊢
          simp only [g, if_false (Nat.succ_ne_zero _),
            Nat.succ_sub_one] at *
          convert hr using 1 <;> norm_num <;> ring

end Submissions.Erdos68TopDigitArbitraryRuns.TopDigitArbitraryRuns
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.Archimedean.Real.Basic

namespace Statements.Erdos68TopDigitArbitraryRuns

/-- The perturbed fractional recurrence admits maximal digits for arbitrarily
long consecutive finite runs. Therefore positivity of the perturbations and
shrinking interval iteration alone cannot yield a universal run bound. -/
abbrev statement : Prop :=
  ∀ M : ℕ, 3 ≤ M → ∀ L : ℕ,
    ∃ f : ℕ → ℝ,
      (∀ i : ℕ, i ≤ L → 0 < f i ∧ f i < 1) ∧
      (∀ i : ℕ, i < L →
        let m := M + i
        ⌊(m : ℝ) * f i + 1 / (m.factorial - 1 : ℕ)⌋ =
            (m - 1 : ℕ) ∧
          f (i + 1) =
            (m : ℝ) * f i + 1 / (m.factorial - 1 : ℕ) -
              (m - 1 : ℕ))

theorem target : statement := sorry

end Statements.Erdos68TopDigitArbitraryRuns
```

### 14. Multiplying old residues by m gives exact quotient carries and residue recurrences; all aggregate carries can…

- Permalink: https://jig.so/p/34?s=14
- Status: kernel-checked
- Filed: 2026-08-25T05:47:10.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Multiplying old residues by m gives exact quotient carries and residue recurrences; all aggregate carries cancel from the finite factorial digit, which is always positive.**

A finite digit at most m−2 survives arbitrary sub-unit tails at both scales.

**Scope.**

For every m≥4, exact old-denominator mod/div recurrences, aggregate A and R recurrences, a closed finite-digit formula and positivity; plus an abstract two-tail survival criterion.

**Artifacts.**

- ResidueCarryRecurrence.lean: Submissions.Erdos68ResidueCarryRecurrence.ResidueCarryRecurrence.combined

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Tactic

open scoped BigOperators

namespace Submissions.Erdos68ResidueCarryRecurrence.ResidueCarryRecurrence

private lemma denominator_pos {n : ℕ} (hn : 2 ≤ n) :
    0 < n.factorial - 1 := by
  have : 1 < n.factorial := Nat.one_lt_factorial.mpr (by omega)
  omega

private lemma mul_mod_recurrence (m M d : ℕ) :
    (m * M) % d = (m * (M % d)) % d := by
  conv_lhs => rw [Nat.mul_mod]
  conv_rhs => rw [Nat.mul_mod]
  rw [Nat.mod_mod]

private lemma mul_div_recurrence (m M d : ℕ) (hd : 0 < d) :
    (m * M) / d =
      m * (M / d) + (m * (M % d)) / d := by
  let k := m * (M / d) + (m * (M % d)) / d
  let r := (m * (M % d)) % d
  have hM := Nat.div_add_mod M d
  have hmr := Nat.div_add_mod (m * (M % d)) d
  have hr : r < d := Nat.mod_lt _ hd
  have hdecomp : k * d + r = m * M := by
    dsimp [k, r] at *
    calc
      (m * (M / d) + m * (M % d) / d) * d +
          m * (M % d) % d =
          m * (d * (M / d)) +
            (d * (m * (M % d) / d) + m * (M % d) % d) := by ring
      _ = m * (d * (M / d)) + m * (M % d) := by rw [hmr]
      _ = m * (d * (M / d) + M % d) := by ring
      _ = m * M := by rw [hM]
  have hlo : k * d ≤ m * M := by omega
  have hhi : m * M < (k + 1) * d := by
    rw [← hdecomp]
    calc
      k * d + r < k * d + d := Nat.add_lt_add_left hr _
      _ = (k + 1) * d := by ring
  apply Nat.div_eq_of_lt_le
  · exact hlo
  · exact hhi

private lemma factorial_step {m : ℕ} (hm : 1 ≤ m) :
    m.factorial = m * (m - 1).factorial :=
  (Nat.mul_factorial_pred (by omega : m ≠ 0)).symm

/-- Exact recurrence for every old denominator: multiplication by `m` updates
the residue by `r ↦ mr mod d` and the quotient by the carry `⌊mr/d⌋`. -/
theorem proof :
    ∀ m : ℕ, 3 ≤ m →
      ∀ n ∈ Finset.Icc 2 (m - 1),
        let d := n.factorial - 1
        let r := (m - 1).factorial % d
        m.factorial % d = (m * r) % d ∧
          m.factorial / d =
            m * ((m - 1).factorial / d) + (m * r) / d := by
  intro m hm n hn
  dsimp
  rw [factorial_step (by omega : 1 ≤ m)]
  constructor
  · exact mul_mod_recurrence _ _ _
  · exact mul_div_recurrence _ _ _
      (denominator_pos (Finset.mem_Icc.mp hn).1)

private lemma div_mod_decomposition (M D : ℕ) (hD : 0 < D) :
    (M : ℝ) / D =
      (M / D : ℕ) + (M % D : ℕ) / (D : ℝ) := by
  have hDR : (D : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hD)
  field_simp [hDR]
  exact_mod_cast (Nat.div_add_mod M D).symm

private lemma scaled_decomposition (k : ℕ) (hk : 2 ≤ k) :
    let A : ℕ :=
      ∑ n ∈ Finset.Icc 2 k, k.factorial / (n.factorial - 1)
    let R : ℝ :=
      ∑ n ∈ Finset.Icc 2 k,
        (k.factorial % (n.factorial - 1) : ℕ) /
          ((n.factorial - 1 : ℕ) : ℝ)
    (k.factorial : ℝ) *
        ∑ n ∈ Finset.Icc 2 k,
          (1 : ℝ) / (n.factorial - 1 : ℕ) = A + R := by
  dsimp
  rw [Finset.mul_sum, Nat.cast_sum, ← Finset.sum_add_distrib]
  apply Finset.sum_congr rfl
  intro n hn
  have hn2 := (Finset.mem_Icc.mp hn).1
  calc
    (k.factorial : ℝ) * ((1 : ℝ) / (n.factorial - 1 : ℕ)) =
        (k.factorial : ℝ) / (n.factorial - 1 : ℕ) := by ring
    _ = (k.factorial / (n.factorial - 1) : ℕ) +
        (k.factorial % (n.factorial - 1) : ℕ) /
          ((n.factorial - 1 : ℕ) : ℝ) :=
      div_mod_decomposition _ _ (denominator_pos hn2)

private lemma floor_radix_step (m : ℕ) (y e : ℝ) :
    ⌊(m : ℝ) * y + 1 + e⌋ - (m : ℤ) * ⌊y⌋ =
      1 + ⌊(m : ℝ) * Int.fract y + e⌋ := by
  have hy : (⌊y⌋ : ℝ) + Int.fract y = y := Int.floor_add_fract y
  have hrearrange :
      (m : ℝ) * y + 1 + e =
        ((m : ℤ) * ⌊y⌋ + 1 : ℤ) +
          ((m : ℝ) * Int.fract y + e) := by
    calc
      (m : ℝ) * y + 1 + e =
          (m : ℝ) * ((⌊y⌋ : ℝ) + Int.fract y) + 1 + e := by rw [hy]
      _ = ((m : ℤ) * ⌊y⌋ + 1 : ℤ) +
          ((m : ℝ) * Int.fract y + e) := by
        push_cast
        ring
  rw [hrearrange, Int.floor_intCast_add]
  omega

-- 253 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Algebra.Order.Archimedean.Real.Basic

open scoped BigOperators

namespace Statements.Erdos68ResidueCarryRecurrence

/-- Exact aggregate carry recurrences, positivity of every finite factorial
digit, and a criterion showing that a finite digit in `[1,m-2]` survives two
positive tails whose scaled sizes are below one. -/
abbrev statement : Prop :=
  (∀ m : ℕ, 4 ≤ m →
    let d : ℕ → ℕ := fun n => n.factorial - 1
    let A : ℕ → ℕ := fun k =>
      ∑ n ∈ Finset.Icc 2 k, k.factorial / d n
    let R : ℕ → ℝ := fun k =>
      ∑ n ∈ Finset.Icc 2 k,
        (k.factorial % d n : ℕ) / (d n : ℝ)
    let C : ℕ :=
      ∑ n ∈ Finset.Icc 2 (m - 1),
        (m * ((m - 1).factorial % d n)) / d n
    let S : ℕ → ℝ := fun k =>
      ∑ n ∈ Finset.Icc 2 k, (1 : ℝ) / d n
    (∀ n ∈ Finset.Icc 2 (m - 1),
      m.factorial % d n =
          (m * ((m - 1).factorial % d n)) % d n ∧
        m.factorial / d n =
          m * ((m - 1).factorial / d n) +
            (m * ((m - 1).factorial % d n)) / d n) ∧
    A m = m * A (m - 1) + C + 1 ∧
    R m =
      (m : ℝ) * R (m - 1) - C +
        1 / (m.factorial - 1 : ℕ) ∧
    ⌊(m.factorial : ℝ) * S m⌋ -
        (m : ℤ) * ⌊((m - 1).factorial : ℝ) * S (m - 1)⌋ =
      1 + ⌊(m : ℝ) *
        Int.fract (((m - 1).factorial : ℝ) * S (m - 1)) +
          1 / (m.factorial - 1 : ℕ)⌋ ∧
    1 ≤
      ⌊(m.factorial : ℝ) * S m⌋ -
        (m : ℤ) * ⌊((m - 1).factorial : ℝ) * S (m - 1)⌋) ∧
  (∀ m : ℕ, 3 ≤ m → ∀ y z u v : ℝ,
    0 ≤ u → u < 1 → 0 ≤ v → v < 1 →
    let D : ℤ := ⌊y⌋ - (m : ℤ) * ⌊z⌋
    1 ≤ D → D ≤ (m : ℤ) - 2 →
    ⌊y + u⌋ - (m : ℤ) * ⌊z + v⌋ ≠ 0)

theorem target : statement := sorry

end Statements.Erdos68ResidueCarryRecurrence
```

### 13. The scaled finite sum m!Sₘ splits into an integer quotient sum and normalized modular residues; its floor spl…

- Permalink: https://jig.so/p/34?s=13
- Status: kernel-checked
- Filed: 2026-08-25T05:33:10.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**The scaled finite sum m!Sₘ splits into an integer quotient sum and normalized modular residues; its floor splits accordingly, and the newest residue is 1.**

**Scope.**

Exact finite formula for every m≥3, rewriting floor(m!Sₘ) through residues m! mod (n!−1).

**Artifacts.**

- FiniteScaledResidueFormula.lean: Submissions.Erdos68FiniteScaledResidueFormula.FiniteScaledResidueFormula.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Tactic

open scoped BigOperators

namespace Submissions.Erdos68FiniteScaledResidueFormula.FiniteScaledResidueFormula

private lemma denominator_pos {n : ℕ} (hn : 2 ≤ n) :
    0 < n.factorial - 1 := by
  have : 1 < n.factorial := Nat.one_lt_factorial.mpr (by omega)
  omega

private lemma div_mod_decomposition (M D : ℕ) (hD : 0 < D) :
    (M : ℝ) / D =
      (M / D : ℕ) + (M % D : ℕ) / (D : ℝ) := by
  have hDR : (D : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hD)
  field_simp [hDR]
  exact_mod_cast (Nat.div_add_mod M D).symm

/-- Scaling the finite sum by `m!` splits it into an explicit integer quotient
sum and a finite sum of normalized modular residues. Consequently its floor is
the quotient sum plus the floor of the residue sum; the newest residue is 1. -/
theorem proof :
    ∀ m : ℕ, 3 ≤ m →
      let A : ℕ :=
        ∑ n ∈ Finset.Icc 2 m,
          m.factorial / (n.factorial - 1)
      let R : ℝ :=
        ∑ n ∈ Finset.Icc 2 m,
          (m.factorial % (n.factorial - 1) : ℕ) /
            ((n.factorial - 1 : ℕ) : ℝ)
      (m.factorial : ℝ) *
          ∑ n ∈ Finset.Icc 2 m,
            (1 : ℝ) / (n.factorial - 1 : ℕ) =
        A + R ∧
      ⌊(m.factorial : ℝ) *
          ∑ n ∈ Finset.Icc 2 m,
            (1 : ℝ) / (n.factorial - 1 : ℕ)⌋ =
        (A : ℤ) + ⌊R⌋ ∧
      m.factorial % (m.factorial - 1) = 1 := by
  intro m hm
  dsimp
  have heq :
      (m.factorial : ℝ) *
          ∑ n ∈ Finset.Icc 2 m,
            (1 : ℝ) / (n.factorial - 1 : ℕ) =
        (∑ n ∈ Finset.Icc 2 m,
          m.factorial / (n.factorial - 1) : ℕ) +
        ∑ n ∈ Finset.Icc 2 m,
          (m.factorial % (n.factorial - 1) : ℕ) /
            ((n.factorial - 1 : ℕ) : ℝ) := by
    rw [Finset.mul_sum, Nat.cast_sum, ← Finset.sum_add_distrib]
    apply Finset.sum_congr rfl
    intro n hn
    have hn2 := (Finset.mem_Icc.mp hn).1
    calc
      (m.factorial : ℝ) *
          ((1 : ℝ) / (n.factorial - 1 : ℕ)) =
          (m.factorial : ℝ) / (n.factorial - 1 : ℕ) := by ring
      _ = (m.factorial / (n.factorial - 1) : ℕ) +
          (m.factorial % (n.factorial - 1) : ℕ) /
            ((n.factorial - 1 : ℕ) : ℝ) :=
        div_mod_decomposition _ _ (denominator_pos hn2)
  refine ⟨heq, ?_, ?_⟩
  · rw [heq]
    rw [Int.floor_natCast_add]
  · have hfac : 1 < m.factorial :=
      Nat.one_lt_factorial.mpr (by omega)
    have hden : 1 < m.factorial - 1 := by
      have hmono := Nat.factorial_le hm
      norm_num at hmono
      omega
    conv_lhs =>
      rw [show m.factorial = (m.factorial - 1) + 1 by omega]
    rw [Nat.add_mod]
    simp [Nat.mod_eq_of_lt hden]

end Submissions.Erdos68FiniteScaledResidueFormula.FiniteScaledResidueFormula
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Algebra.Order.Archimedean.Real.Basic

open scoped BigOperators

namespace Statements.Erdos68FiniteScaledResidueFormula

/-- Scaling the finite sum by `m!` splits it into an explicit integer quotient
sum and a finite sum of normalized modular residues. Consequently its floor is
the quotient sum plus the floor of the residue sum; the newest residue is 1. -/
abbrev statement : Prop :=
  ∀ m : ℕ, 3 ≤ m →
    let A : ℕ :=
      ∑ n ∈ Finset.Icc 2 m,
        m.factorial / (n.factorial - 1)
    let R : ℝ :=
      ∑ n ∈ Finset.Icc 2 m,
        (m.factorial % (n.factorial - 1) : ℕ) /
          ((n.factorial - 1 : ℕ) : ℝ)
    (m.factorial : ℝ) *
        ∑ n ∈ Finset.Icc 2 m,
          (1 : ℝ) / (n.factorial - 1 : ℕ) =
      A + R ∧
    ⌊(m.factorial : ℝ) *
        ∑ n ∈ Finset.Icc 2 m,
          (1 : ℝ) / (n.factorial - 1 : ℕ)⌋ =
      (A : ℤ) + ⌊R⌋ ∧
    m.factorial % (m.factorial - 1) = 1

theorem target : statement := sorry

end Statements.Erdos68FiniteScaledResidueFormula
```

### 12. At m=4, the first omitted row contributes less than one scaled unit but changes floor(4!·S₄) from 29 to 30.

- Permalink: https://jig.so/p/34?s=12
- Status: dead route
- Filed: 2026-08-25T05:29:46.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**At m=4, the first omitted row contributes less than one scaled unit but changes floor(4!·S₄) from 29 to 30.**

**Scope.**

Exact rational counterexample using S₄ and the n=5 row; no approximation or full-tail computation.

**Artifacts.**

- FloorStabilityCounterexample.lean: Submissions.Erdos68FloorStabilityCounterexample.FloorStabilityCounterexample.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Data.Real.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68FloorStabilityCounterexample.FloorStabilityCounterexample

/-- Although the first omitted contribution scaled by `4!` is below one, it
already changes the floor of the scaled finite sum. -/
theorem proof :
    let S4 : ℝ :=
      1 / ((2 : ℕ).factorial - 1 : ℕ) +
      1 / ((3 : ℕ).factorial - 1 : ℕ) +
      1 / ((4 : ℕ).factorial - 1 : ℕ)
    let t : ℝ :=
      (4 : ℕ).factorial / ((5 : ℕ).factorial - 1 : ℕ)
    ⌊((4 : ℕ).factorial : ℝ) * S4⌋ = 29 ∧
      ⌊((4 : ℕ).factorial : ℝ) * S4 + t⌋ = 30 ∧
      0 < t ∧ t < 1 := by
  norm_num [Nat.factorial, Int.floor]

end Submissions.Erdos68FloorStabilityCounterexample.FloorStabilityCounterexample
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Data.Real.Basic

namespace Statements.Erdos68FloorStabilityCounterexample

/-- Although the first omitted contribution scaled by `4!` is below one, it
already changes the floor of the scaled finite sum. -/
abbrev statement : Prop :=
  let S4 : ℝ :=
    1 / ((2 : ℕ).factorial - 1 : ℕ) +
    1 / ((3 : ℕ).factorial - 1 : ℕ) +
    1 / ((4 : ℕ).factorial - 1 : ℕ)
  let t : ℝ :=
    (4 : ℕ).factorial / ((5 : ℕ).factorial - 1 : ℕ)
  ⌊((4 : ℕ).factorial : ℝ) * S4⌋ = 29 ∧
    ⌊((4 : ℕ).factorial : ℝ) * S4 + t⌋ = 30 ∧
    0 < t ∧ t < 1

theorem target : statement := sorry

end Statements.Erdos68FloorStabilityCounterexample
```

### 11. The canonical digit floor(m!x)−m floor((m−1)!x) lies in [0,m); floor stability requires the tail to lie below…

- Permalink: https://jig.so/p/34?s=11
- Status: kernel-checked
- Filed: 2026-08-25T05:29:35.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**The canonical digit floor(m!x)−m floor((m−1)!x) lies in [0,m); floor stability requires the tail to lie below the actual fractional margin; an integral preceding scale forces digit zero.**

**Scope.**

General real factorial-digit bounds, the correct positive-tail floor-stability condition, and the integral-scale vanishing criterion.

**Artifacts.**

- FactorialDigitFormula.lean: Submissions.Erdos68FactorialDigitFormula.FactorialDigitFormula.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Data.Real.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68FactorialDigitFormula.FactorialDigitFormula

private lemma floor_stable_below_margin (y t : ℝ) (ht0 : 0 ≤ t)
    (ht : t < (⌊y⌋ : ℝ) + 1 - y) :
    ⌊y + t⌋ = ⌊y⌋ := by
  rw [Int.floor_eq_iff]
  constructor
  · exact (Int.floor_le y).trans (le_add_of_nonneg_right ht0)
  · linarith

/-- The canonical factorial digit lies in its radix range; a positive tail
preserves the relevant floor precisely when it is below the fractional margin;
and an already-integral preceding scale forces the digit to vanish. -/
theorem proof :
    (∀ m : ℕ, 1 ≤ m → ∀ x : ℝ,
      let a : ℤ :=
        ⌊(m.factorial : ℝ) * x⌋ -
          (m : ℤ) * ⌊((m - 1).factorial : ℝ) * x⌋
      0 ≤ a ∧ a < m) ∧
    (∀ y t : ℝ, 0 ≤ t →
      t < (⌊y⌋ : ℝ) + 1 - y →
      ⌊y + t⌋ = ⌊y⌋) ∧
    (∀ m : ℕ, 1 ≤ m → ∀ x : ℝ, ∀ z : ℤ,
      ((m - 1).factorial : ℝ) * x = z →
      ⌊(m.factorial : ℝ) * x⌋ -
          (m : ℤ) * ⌊((m - 1).factorial : ℝ) * x⌋ = 0) := by
  refine ⟨?_, ?_, ?_⟩
  · intro m hm x
    dsimp
    have hfac : m.factorial = m * (m - 1).factorial :=
      (Nat.mul_factorial_pred (by omega : m ≠ 0)).symm
    let y : ℝ := ((m - 1).factorial : ℝ) * x
    have hm0 : (0 : ℝ) ≤ m := by positivity
    have hlowerR :
        (m : ℝ) * (⌊y⌋ : ℝ) ≤ (m : ℝ) * y :=
      mul_le_mul_of_nonneg_left (Int.floor_le y) hm0
    have hlower :
        (m : ℤ) * ⌊y⌋ ≤ ⌊(m : ℝ) * y⌋ := by
      rw [Int.le_floor]
      exact_mod_cast hlowerR
    have hupperR :
        (m : ℝ) * y <
          ((m : ℤ) * (⌊y⌋ + 1) : ℤ) := by
      push_cast
      exact mul_lt_mul_of_pos_left (Int.lt_floor_add_one y)
        (by exact_mod_cast (by omega : 0 < m))
    have hupper :
        ⌊(m : ℝ) * y⌋ < (m : ℤ) * (⌊y⌋ + 1) := by
      rw [Int.floor_lt]
      exact hupperR
    have hscaled :
        (m.factorial : ℝ) * x = (m : ℝ) * y := by
      rw [hfac, Nat.cast_mul]
      dsimp [y]
      ring
    have hprev :
        ((m - 1).factorial : ℝ) * x = y := rfl
    rw [hscaled, hprev]
    rw [mul_add, mul_one] at hupper
    constructor <;> omega
  · exact floor_stable_below_margin
  · intro m hm x z hz
    have hfac : m.factorial = m * (m - 1).factorial :=
      (Nat.mul_factorial_pred (by omega : m ≠ 0)).symm
    have hscaled :
        (m.factorial : ℝ) * x = (((m : ℤ) * z : ℤ) : ℝ) := by
      rw [hfac, Nat.cast_mul, mul_assoc, hz]
      norm_num
    have hprevFloor :
        ⌊((m - 1).factorial : ℝ) * x⌋ = z := by
      rw [hz]
      exact Int.floor_intCast z
    rw [hscaled, hprevFloor, Int.floor_intCast]
    omega

end Submissions.Erdos68FactorialDigitFormula.FactorialDigitFormula
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Data.Real.Basic

namespace Statements.Erdos68FactorialDigitFormula

/-- The canonical factorial digit lies in its radix range; a positive tail
preserves the relevant floor when it is below the fractional margin; and an
already-integral preceding scale forces the digit to vanish. -/
abbrev statement : Prop :=
  (∀ m : ℕ, 1 ≤ m → ∀ x : ℝ,
    let a : ℤ :=
      ⌊(m.factorial : ℝ) * x⌋ -
        (m : ℤ) * ⌊((m - 1).factorial : ℝ) * x⌋
    0 ≤ a ∧ a < m) ∧
  (∀ y t : ℝ, 0 ≤ t →
    t < (⌊y⌋ : ℝ) + 1 - y →
    ⌊y + t⌋ = ⌊y⌋) ∧
  (∀ m : ℕ, 1 ≤ m → ∀ x : ℝ, ∀ z : ℤ,
    ((m - 1).factorial : ℝ) * x = z →
    ⌊(m.factorial : ℝ) * x⌋ -
        (m : ℤ) * ⌊((m - 1).factorial : ℝ) * x⌋ = 0)

theorem target : statement := sorry

end Statements.Erdos68FactorialDigitFormula
```

### 10. For m≥2, the complete omitted tail after row m is summable and, after multiplication by m!, is at most (m+1)/…

- Permalink: https://jig.so/p/34?s=10
- Status: kernel-checked
- Filed: 2026-08-25T05:29:23.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**For m≥2, the complete omitted tail after row m is summable and, after multiplication by m!, is at most (m+1)/m²<1.**

**Scope.**

Uniform analytic bound on the full positive outer tail Σ_{k≥0}1/((m+k+1)!−1), including summability and nonnegativity.

**Artifacts.**

- FactorialTailBound.lean: Submissions.Erdos68FactorialTailBound.FactorialTailBound.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68FactorialTailBound.FactorialTailBound

private lemma denominator_lower (m k : ℕ) (hm : 2 ≤ m) :
    m * m.factorial * (m + 1) ^ k ≤
      (m + k + 1).factorial - 1 := by
  have hA0 : 0 < m.factorial * (m + 1) ^ k := by positivity
  have hA : 1 ≤ m.factorial * (m + 1) ^ k := by omega
  have hstep :
      m * m.factorial * (m + 1) ^ k + 1 ≤
        m.factorial * (m + 1) ^ (k + 1) := by
    calc
      m * m.factorial * (m + 1) ^ k + 1
          ≤ m * m.factorial * (m + 1) ^ k +
              m.factorial * (m + 1) ^ k :=
        Nat.add_le_add_left hA _
      _ = m.factorial * (m + 1) ^ (k + 1) := by
        rw [pow_succ]
        ring
  have hfac :
      m.factorial * (m + 1) ^ (k + 1) ≤
        (m + k + 1).factorial := by
    simpa [Nat.add_assoc] using
      (@Nat.factorial_mul_pow_le_factorial m (k + 1))
  omega

private lemma term_bound (m k : ℕ) (hm : 2 ≤ m) :
    (1 : ℝ) / (((m + k + 1).factorial - 1 : ℕ) : ℝ) ≤
      (1 / ((m : ℝ) * m.factorial)) *
        (1 / ((m : ℝ) + 1)) ^ k := by
  have hL : 0 < m * m.factorial * (m + 1) ^ k := by positivity
  have hle := denominator_lower m k hm
  have hleR :
      ((m * m.factorial * (m + 1) ^ k : ℕ) : ℝ) ≤
        (((m + k + 1).factorial - 1 : ℕ) : ℝ) := by
    exact_mod_cast hle
  calc
    (1 : ℝ) / (((m + k + 1).factorial - 1 : ℕ) : ℝ)
        ≤ 1 / ((m * m.factorial * (m + 1) ^ k : ℕ) : ℝ) :=
      one_div_le_one_div_of_le (by exact_mod_cast hL) hleR
    _ = (1 / ((m : ℝ) * m.factorial)) *
          (1 / ((m : ℝ) + 1)) ^ k := by
      push_cast
      rw [div_pow]
      field_simp
      simp

/-- The complete outer tail after row `m`, when scaled by `m!`, is at most
`(m+1)/m²`, hence strictly less than one. -/
theorem proof :
    ∀ m : ℕ, 2 ≤ m →
      let f : ℕ → ℝ := fun k =>
        1 / (((m + k + 1).factorial - 1 : ℕ) : ℝ)
      Summable f ∧
        0 ≤ (m.factorial : ℝ) * ∑' k, f k ∧
        (m.factorial : ℝ) * ∑' k, f k ≤
          (m + 1 : ℝ) / m ^ 2 ∧
        (m + 1 : ℝ) / m ^ 2 < 1 := by
  intro m hm
  dsimp
  let C : ℝ := 1 / ((m : ℝ) * m.factorial)
  let r : ℝ := 1 / ((m : ℝ) + 1)
  have hr0 : 0 ≤ r := by dsimp [r]; positivity
  have hr1 : r < 1 := by
    dsimp [r]
    have h : (1 : ℝ) < (m : ℝ) + 1 := by
      exact_mod_cast (by omega : 1 < m + 1)
    simpa using one_div_lt_one_div_of_lt (by norm_num : (0 : ℝ) < 1) h
  have hg : Summable (fun k : ℕ => C * r ^ k) :=
    (summable_geometric_of_lt_one hr0 hr1).mul_left C
  have hf :
      Summable (fun k : ℕ =>
        (1 : ℝ) / (((m + k + 1).factorial - 1 : ℕ) : ℝ)) := by
    exact Summable.of_nonneg_of_le
      (fun k => by positivity)
      (fun k => by simpa only [C, r] using term_bound m k hm)
      hg
  refine ⟨hf, ?_, ?_, ?_⟩
  · positivity
  · have hle := hf.tsum_le_tsum
        (fun k => by simpa only [C, r] using term_bound m k hm) hg
    rw [tsum_mul_left, tsum_geometric_of_lt_one hr0 hr1] at hle
    dsimp [C, r] at hle
    have hmR : (0 : ℝ) < m := by exact_mod_cast (by omega : 0 < m)
    have hfacR : (0 : ℝ) < m.factorial := by positivity
    calc
      (m.factorial : ℝ) *
          ∑' k : ℕ, 1 / (↑((m + k + 1).factorial - 1) : ℝ)
          ≤ (m.factorial : ℝ) *
              ((1 / ((m : ℝ) * m.factorial)) *
                (1 - 1 / ((m : ℝ) + 1))⁻¹) :=
        mul_le_mul_of_nonneg_left hle (by positivity)
      _ = (m + 1 : ℝ) / m ^ 2 := by
        field_simp [ne_of_gt hmR, ne_of_gt hfacR]
        ring
  · have hmR : (2 : ℝ) ≤ m := by exact_mod_cast hm
    push_cast
    rw [div_lt_one (by positivity : (0 : ℝ) < m ^ 2)]
    nlinarith

end Submissions.Erdos68FactorialTailBound.FactorialTailBound
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Data.Real.Basic

namespace Statements.Erdos68FactorialTailBound

/-- The complete outer tail after row `m`, when scaled by `m!`, is at most
`(m+1)/m²`, hence strictly less than one. -/
abbrev statement : Prop :=
  ∀ m : ℕ, 2 ≤ m →
    let f : ℕ → ℝ := fun k =>
      1 / (((m + k + 1).factorial - 1 : ℕ) : ℝ)
    Summable f ∧
      0 ≤ (m.factorial : ℝ) * ∑' k, f k ∧
      (m.factorial : ℝ) * ∑' k, f k ≤
        (m + 1 : ℝ) / m ^ 2 ∧
      (m + 1 : ℝ) / m ^ 2 < 1

theorem target : statement := sorry

end Statements.Erdos68FactorialTailBound
```

### 9. If m lies beyond the explicit finite-row termination position K!, then the first omitted row 1/((K+1)!−1) is…

- Permalink: https://jig.so/p/34?s=9
- Status: dead route
- Filed: 2026-08-25T05:17:25.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**If m lies beyond the explicit finite-row termination position K!, then the first omitted row 1/((K+1)!−1) is already larger than one full 1/m!**

Unit.

**Scope.**

For all K ≥ 3 and m > K!, the reciprocal at factorial position m is strictly smaller than the first row omitted after K.

**Artifacts.**

- FiniteRowTailObstruction.lean: Submissions.Erdos68FiniteRowTailObstruction.FiniteRowTailObstruction.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68FiniteRowTailObstruction.FiniteRowTailObstruction

/-- Using the explicit common termination position `K!`, every later target
position still has a first omitted row larger than one whole target unit. -/
theorem proof :
    ∀ K m : ℕ, 3 ≤ K → K.factorial < m →
      (1 : ℝ) / m.factorial <
        1 / (((K + 1).factorial - 1 : ℕ) : ℝ) := by
  intro K m hK hm
  have hKfac : K ≤ K.factorial := Nat.self_le_factorial K
  have hKm : K + 1 ≤ m := by omega
  have hfacLe : (K + 1).factorial ≤ m.factorial :=
    Nat.factorial_le hKm
  have hdenPos : 0 < (K + 1).factorial - 1 := by
    have : 1 < (K + 1).factorial :=
      Nat.one_lt_factorial.mpr (by omega)
    omega
  have hdenLt : (K + 1).factorial - 1 < m.factorial := by omega
  have hdenLtR :
      ((((K + 1).factorial - 1 : ℕ) : ℝ)) <
        (m.factorial : ℝ) := by
    exact_mod_cast hdenLt
  exact one_div_lt_one_div_of_lt (by exact_mod_cast hdenPos) hdenLtR

end Submissions.Erdos68FiniteRowTailObstruction.FiniteRowTailObstruction
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Real.Basic

namespace Statements.Erdos68FiniteRowTailObstruction

/-- Using the explicit common termination position `K!`, every later target
position still has a first omitted row larger than one whole target unit. -/
abbrev statement : Prop :=
  ∀ K m : ℕ, 3 ≤ K → K.factorial < m →
    (1 : ℝ) / m.factorial <
      1 / (((K + 1).factorial - 1 : ℕ) : ℝ)

theorem target : statement := sorry

end Statements.Erdos68FiniteRowTailObstruction
```

### 8. Each rational row 1/(n!−1) terminates by factorial position n!−1, and all rows through K admit one explicit c…

- Permalink: https://jig.so/p/34?s=8
- Status: kernel-checked
- Filed: 2026-08-25T05:13:32.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Each rational row 1/(n!−1) terminates by factorial position n!−1, and all rows through K admit one explicit common denominator (K!)!.**

**Scope.**

For every n ≥ 2, divisibility of n!−1 into (n!−1)!; and for every K ≥ 2, an exact rational representation of the rows 2 through K over (K!)!.

**Artifacts.**

- FiniteRowTermination.lean: Submissions.Erdos68FiniteRowTermination.FiniteRowTermination.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.Rat.Defs
import Mathlib.Tactic

open scoped BigOperators

namespace Submissions.Erdos68FiniteRowTermination.FiniteRowTermination

private lemma denominator_pos {n : ℕ} (hn : 2 ≤ n) :
    0 < n.factorial - 1 := by
  have hfac : 1 < n.factorial := Nat.one_lt_factorial.mpr (by omega)
  omega

private lemma denominator_dvd_common_factorial {n K : ℕ}
    (hn : 2 ≤ n) (hnK : n ≤ K) :
    n.factorial - 1 ∣ K.factorial.factorial := by
  apply Nat.dvd_factorial (denominator_pos hn)
  exact le_trans (Nat.sub_le _ _) (Nat.factorial_le hnK)

/-- Every single row has an explicit terminating factorial denominator, and all
rows through `K` share the computable factorial position `K!`. -/
theorem proof :
    (∀ n : ℕ, 2 ≤ n →
      let d := n.factorial - 1
      d ∣ d.factorial) ∧
    ∀ K : ℕ, 2 ≤ K →
      let Q := K.factorial.factorial
      let A :=
        ∑ n ∈ Finset.Icc 2 K, Q / (n.factorial - 1)
      (∑ n ∈ Finset.Icc 2 K,
          (1 : ℚ) / (n.factorial - 1 : ℕ)) =
        (A : ℚ) / Q := by
  constructor
  · intro n hn
    dsimp
    exact Nat.dvd_factorial (denominator_pos hn) le_rfl
  · intro K hK
    dsimp
    rw [Nat.cast_sum, Finset.sum_div]
    apply Finset.sum_congr rfl
    intro n hn
    have hnIcc := Finset.mem_Icc.mp hn
    have hdvd := denominator_dvd_common_factorial hnIcc.1 hnIcc.2
    have hd0 : ((n.factorial - 1 : ℕ) : ℚ) ≠ 0 := by
      exact_mod_cast (ne_of_gt (denominator_pos hnIcc.1))
    have hQ0 : ((K.factorial.factorial : ℕ) : ℚ) ≠ 0 := by
      exact_mod_cast Nat.factorial_ne_zero K.factorial
    apply (div_eq_div_iff hd0 hQ0).2
    norm_num
    exact_mod_cast (Nat.div_mul_cancel hdvd).symm

end Submissions.Erdos68FiniteRowTermination.FiniteRowTermination
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.Rat.Defs

open scoped BigOperators

namespace Statements.Erdos68FiniteRowTermination

/-- Every single row has an explicit terminating factorial denominator, and all
rows through `K` share the computable factorial position `K!`. -/
abbrev statement : Prop :=
  (∀ n : ℕ, 2 ≤ n →
    let d := n.factorial - 1
    d ∣ d.factorial) ∧
  ∀ K : ℕ, 2 ≤ K →
    let Q := K.factorial.factorial
    let A :=
      ∑ n ∈ Finset.Icc 2 K, Q / (n.factorial - 1)
    (∑ n ∈ Finset.Icc 2 K,
        (1 : ℚ) / (n.factorial - 1 : ℕ)) =
      (A : ℚ) / Q

theorem target : statement := sorry

end Statements.Erdos68FiniteRowTermination
```

### 7. At every prime factorial position m ≥ 5, the n=3 geometric row contains a term assigned above m whose size al…

- Permalink: https://jig.so/p/34?s=7
- Status: dead route
- Filed: 2026-08-25T05:07:03.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**At every prime factorial position m ≥ 5, the n=3 geometric row contains a term assigned above m whose size already exceeds one full 1/m!**

Unit.

**Scope.**

For every prime m ≥ 5, with j=⌊m/3⌋+1, the raw pair (3,j) lies above position m and 6^(−j) > 1/m!.

**Artifacts.**

- PrimePositionCarryObstruction.lean: Submissions.Erdos68PrimePositionCarryObstruction.PrimePositionCarryObstruction.proof

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Tactic

namespace Submissions.Erdos68PrimePositionCarryObstruction.PrimePositionCarryObstruction

private lemma six_pow_lt_factorial_shift :
    ∀ t : ℕ, 6 ^ ((t + 5) / 3 + 1) < (t + 5).factorial := by
  intro t
  induction t with
  | zero => norm_num [Nat.factorial]
  | succ t ih =>
      have hdiv :
          (t + 1 + 5) / 3 + 1 ≤ ((t + 5) / 3 + 1) + 1 := by omega
      calc
        6 ^ ((t + 1 + 5) / 3 + 1)
            ≤ 6 ^ (((t + 5) / 3 + 1) + 1) :=
          Nat.pow_le_pow_right (by norm_num) hdiv
        _ = 6 ^ ((t + 5) / 3 + 1) * 6 := by rw [pow_succ]
        _ < (t + 5).factorial * 6 :=
          Nat.mul_lt_mul_of_pos_right ih (by norm_num)
        _ ≤ (t + 5).factorial * (t + 6) := by
          apply Nat.mul_le_mul_left
          omega
        _ = (t + 1 + 5).factorial := by
          conv_rhs =>
            rw [show t + 1 + 5 = (t + 5) + 1 by omega, Nat.factorial_succ]
          rw [Nat.mul_comm]

private lemma six_pow_lt_factorial {m : ℕ} (hm : 5 ≤ m) :
    6 ^ (m / 3 + 1) < m.factorial := by
  obtain ⟨t, rfl⟩ := Nat.exists_eq_add_of_le hm
  simpa [Nat.add_comm] using six_pow_lt_factorial_shift t

/-- At every prime factorial position `m ≥ 5`, a single term from the `n = 3`
geometric row is assigned to a higher raw position but is already larger than
one full `1 / m!` unit. Thus the unnormalized higher-position tail cannot be
bounded below one unit at `m`. -/
theorem proof :
    ∀ m : ℕ, m.Prime → 5 ≤ m →
      let j := m / 3 + 1
      m < 3 * j ∧
        (1 : ℝ) / m.factorial < 1 / (6 : ℝ) ^ j := by
  intro m hmprime hm
  dsimp
  constructor
  · omega
  · have hpow := six_pow_lt_factorial hm
    have hpowR : (6 : ℝ) ^ (m / 3 + 1) < (m.factorial : ℝ) := by
      exact_mod_cast hpow
    have hleft : 0 < (6 : ℝ) ^ (m / 3 + 1) := by positivity
    have hright : 0 < (m.factorial : ℝ) := by positivity
    exact one_div_lt_one_div_of_lt hleft hpowR

end Submissions.Erdos68PrimePositionCarryObstruction.PrimePositionCarryObstruction
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Real.Basic

namespace Statements.Erdos68PrimePositionCarryObstruction

/-- At every prime factorial position `m ≥ 5`, a single term from the `n = 3`
geometric row is assigned to a higher raw position but is already larger than
one full `1 / m!` unit. Thus the unnormalized higher-position tail cannot be
bounded below one unit at `m`. -/
abbrev statement : Prop :=
  ∀ m : ℕ, m.Prime → 5 ≤ m →
    let j := m / 3 + 1
    m < 3 * j ∧
      (1 : ℝ) / m.factorial < 1 / (6 : ℝ) ^ j

theorem target : statement := sorry

end Statements.Erdos68PrimePositionCarryObstruction
```

### 6. For n ≥ 3, truncating the repeating base-n!

- Permalink: https://jig.so/p/34?s=6
- Status: kernel-checked
- Filed: 2026-08-25T04:58:37.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**For n ≥ 3, truncating the repeating base-n!**

Expansion of 1/(n!−1) after J digits leaves an exact remainder strictly between one and two units at the next base-n! position.

**Scope.**

For every natural n ≥ 3 and truncation length J ≥ 0, exact identity and strict next-position bounds for the finite geometric remainder.

**Artifacts.**

- GeometricCarry.lean: Submissions.Erdos68GeometricCarry.GeometricCarry.proof

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Real.Basic
import Mathlib.Tactic

open scoped BigOperators

namespace Submissions.Erdos68GeometricCarry.GeometricCarry

private lemma finite_geometric_remainder (B : ℝ) (hB0 : B ≠ 0)
    (hB1 : B ≠ 1) :
    ∀ J : ℕ,
      1 / (B - 1) - ∑ j ∈ Finset.range J, 1 / B ^ (j + 1) =
        1 / (B ^ J * (B - 1)) := by
  intro J
  induction J with
  | zero => simp
  | succ J ih =>
      rw [Finset.sum_range_succ, sub_add_eq_sub_sub, ih]
      field_simp
      ring

/-- After truncating the repeating base-`n!` expansion of `1 / (n! - 1)`
after `J` digits, the remainder is exactly known and lies strictly between one
and two units at the next base-`n!` position. -/
theorem proof :
    ∀ n J : ℕ, 3 ≤ n →
      let B : ℝ := n.factorial
      let R : ℝ :=
        1 / (B - 1) - ∑ j ∈ Finset.range J, 1 / B ^ (j + 1)
      R = 1 / (B ^ J * (B - 1)) ∧
        1 / B ^ (J + 1) < R ∧
        R < 2 / B ^ (J + 1) := by
  intro n J hn
  dsimp
  have hfacNat : 2 < n.factorial := by
    have hmono := Nat.factorial_le hn
    norm_num at hmono ⊢
    omega
  have hB : (2 : ℝ) < (n.factorial : ℝ) := by exact_mod_cast hfacNat
  have hB0 : (n.factorial : ℝ) ≠ 0 := by positivity
  have hB1 : (n.factorial : ℝ) ≠ 1 := by linarith
  have hpow : 0 < (n.factorial : ℝ) ^ J := by positivity
  have hpow' : 0 < (n.factorial : ℝ) ^ (J + 1) := by positivity
  have hsub : 0 < (n.factorial : ℝ) - 1 := by linarith
  have heq := finite_geometric_remainder (n.factorial : ℝ) hB0 hB1 J
  refine ⟨heq, ?_, ?_⟩
  · rw [heq]
    rw [div_lt_div_iff₀ hpow' (mul_pos hpow hsub)]
    rw [pow_succ]
    nlinarith
  · rw [heq]
    rw [div_lt_div_iff₀ (mul_pos hpow hsub) hpow']
    rw [pow_succ]
    nlinarith

end Submissions.Erdos68GeometricCarry.GeometricCarry
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Real.Basic

open scoped BigOperators

namespace Statements.Erdos68GeometricCarry

/-- After truncating the repeating base-`n!` expansion of `1 / (n! - 1)`
after `J` digits, the remainder is exactly known and lies strictly between one
and two units at the next base-`n!` position. -/
abbrev statement : Prop :=
  ∀ n J : ℕ, 3 ≤ n →
    let B : ℝ := n.factorial
    let R : ℝ :=
      1 / (B - 1) - ∑ j ∈ Finset.range J, 1 / B ^ (j + 1)
    R = 1 / (B ^ J * (B - 1)) ∧
      1 / B ^ (J + 1) < R ∧
      R < 2 / B ^ (J + 1)

theorem target : statement := sorry

end Statements.Erdos68GeometricCarry
```

### 5. If one denominator k!−1 in the finite sum through N has strictly largest p-adic valuation, then that reciproc…

- Permalink: https://jig.so/p/34?s=5
- Status: kernel-checked
- Filed: 2026-08-25T04:50:49.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**If one denominator k!−1 in the finite sum through N has strictly largest p-adic valuation, then that reciprocal uniquely controls the p-adic valuation of the entire rational sum.**

**Scope.**

For all N ≥ k ≥ 2 and primes p, assuming k!−1 has strictly greater p-adic valuation than every other n!−1 with 2 ≤ n ≤ N.

**Artifacts.**

- UniqueMaxValuation.lean: Submissions.Erdos68UniqueMaxValuation.UniqueMaxValuation.proof

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic

open scoped BigOperators

namespace Submissions.Erdos68UniqueMaxValuation.UniqueMaxValuation

private theorem unique_min_sum
    {ι : Type*} [DecidableEq ι] (p : ℕ) [Fact p.Prime]
    (t : Finset ι) (f : ι → ℚ) (k : ι)
    (hk0 : f k ≠ 0)
    (ht0 : ∀ i ∈ t, f i ≠ 0)
    (hmin : ∀ i ∈ t, padicValRat p (f k) < padicValRat p (f i)) :
    (∑ i ∈ t, f i) + f k ≠ 0 ∧
      padicValRat p (∑ i ∈ t, f i + f k) = padicValRat p (f k) := by
  induction t using Finset.induction with
  | empty => simp [hk0]
  | @insert a t ha ih =>
      have hi := ih (fun i hi => ht0 i (Finset.mem_insert_of_mem hi))
        (fun i hi => hmin i (Finset.mem_insert_of_mem hi))
      have hsum0 : (∑ i ∈ t, f i) + f k ≠ 0 := hi.1
      have hne :
          padicValRat p ((∑ i ∈ t, f i) + f k) ≠ padicValRat p (f a) := by
        rw [hi.2]
        exact ne_of_lt (hmin a (Finset.mem_insert_self a t))
      have hadd0 : ((∑ i ∈ t, f i) + f k) + f a ≠ 0 := by
        intro hzero
        have heq := eq_neg_of_add_eq_zero_left hzero
        have hv := congrArg (padicValRat p) heq
        rw [padicValRat.neg] at hv
        exact hne hv
      have hadd := padicValRat.add_eq_min hadd0 hsum0
        (ht0 a (Finset.mem_insert_self a t)) hne
      have heq :
          f a + (∑ i ∈ t, f i) + f k =
            ((∑ i ∈ t, f i) + f k) + f a := by ring
      constructor
      · rw [Finset.sum_insert ha, heq]
        exact hadd0
      · rw [Finset.sum_insert ha, heq, hadd, min_eq_left, hi.2]
        rw [hi.2]
        exact (hmin a (Finset.mem_insert_self a t)).le

private lemma factorial_sub_one_ne_zero {n : ℕ} (hn : 2 ≤ n) :
    n.factorial - 1 ≠ 0 := by
  have hfac : 1 < n.factorial := Nat.one_lt_factorial.mpr (by omega)
  omega

private lemma reciprocal_valuation (p n : ℕ) [Fact p.Prime] (hn : 2 ≤ n) :
    padicValRat p ((1 : ℚ) / (n.factorial - 1 : ℕ)) =
      -(padicValNat p (n.factorial - 1) : ℤ) := by
  have hdenNat := factorial_sub_one_ne_zero hn
  have hdenRat : ((n.factorial - 1 : ℕ) : ℚ) ≠ 0 := by exact_mod_cast hdenNat
  rw [padicValRat.div one_ne_zero hdenRat, padicValRat.one, ← padicValRat_of_nat]
  omega

/-- If one factorial-minus-one denominator in a finite truncation has strictly
largest `p`-adic valuation, its reciprocal has uniquely smallest valuation and
therefore determines the valuation of the whole finite sum. -/
theorem proof :
    ∀ N k p : ℕ, 2 ≤ k → k ≤ N → p.Prime →
      (∀ n ∈ Finset.Icc 2 N, n ≠ k →
        padicValNat p (n.factorial - 1) <
          padicValNat p (k.factorial - 1)) →
      padicValRat p
          (∑ n ∈ Finset.Icc 2 N,
            (1 : ℚ) / (n.factorial - 1 : ℕ)) =
        -(padicValNat p (k.factorial - 1) : ℤ) := by
  intro N k p hk hNk hp hmax
  letI : Fact p.Prime := ⟨hp⟩
  let s := Finset.Icc 2 N
  let q : ℕ → ℚ := fun n => (1 : ℚ) / (n.factorial - 1 : ℕ)
  have hks : k ∈ s := Finset.mem_Icc.mpr ⟨hk, hNk⟩
  have hqk0 : q k ≠ 0 := by
    apply div_ne_zero one_ne_zero
    exact_mod_cast factorial_sub_one_ne_zero hk
  have hqi0 : ∀ n ∈ s.erase k, q n ≠ 0 := by
    intro n hn
    apply div_ne_zero one_ne_zero
    exact_mod_cast factorial_sub_one_ne_zero (Finset.mem_Icc.mp (Finset.mem_of_mem_erase hn)).1
  have hqmin :
      ∀ n ∈ s.erase k, padicValRat p (q k) < padicValRat p (q n) := by
    intro n hn
    have hns := Finset.mem_of_mem_erase hn
    have hnk := Finset.ne_of_mem_erase hn
    rw [reciprocal_valuation p k hk,
      reciprocal_valuation p n (Finset.mem_Icc.mp hns).1]
    exact neg_lt_neg (mod_cast hmax n hns hnk)
  rw [← Finset.sum_erase_add s q hks]
  calc
    padicValRat p (∑ n ∈ s.erase k, q n + q k) =
        padicValRat p (q k) :=
      (unique_min_sum p (s.erase k) q k hqk0 hqi0 hqmin).2
    _ = -(padicValNat p (k.factorial - 1) : ℤ) :=
      reciprocal_valuation p k hk

end Submissions.Erdos68UniqueMaxValuation.UniqueMaxValuation
```

- Canonical statement

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat

open scoped BigOperators

namespace Statements.Erdos68UniqueMaxValuation

/-- If one factorial-minus-one denominator in a finite truncation has strictly
largest `p`-adic valuation, its reciprocal has uniquely smallest valuation and
therefore determines the valuation of the whole finite sum. -/
abbrev statement : Prop :=
  ∀ N k p : ℕ, 2 ≤ k → k ≤ N → p.Prime →
    (∀ n ∈ Finset.Icc 2 N, n ≠ k →
      padicValNat p (n.factorial - 1) <
        padicValNat p (k.factorial - 1)) →
    padicValRat p
        (∑ n ∈ Finset.Icc 2 N,
          (1 : ℚ) / (n.factorial - 1 : ℕ)) =
      -(padicValNat p (k.factorial - 1) : ℤ)

theorem target : statement := sorry

end Statements.Erdos68UniqueMaxValuation
```

### 4. If a prime p divides N!−1 but no earlier n!−1 for 2 ≤ n < N, then p does not divide the canonical common nume…

- Permalink: https://jig.so/p/34?s=4
- Status: kernel-checked
- Filed: 2026-08-25T04:40:10.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**If a prime p divides N!−1 but no earlier n!−1 for 2 ≤ n < N, then p does not divide the canonical common numerator of the finite reciprocal sum through N.**

**Scope.**

For every N ≥ 2 and prime p that divides N!−1 but no n!−1 with 2 ≤ n < N, nondivisibility of p from the finite sum common numerator.

**Artifacts.**

- PrimitiveNonCancellation.lean: Submissions.Erdos68PrimitiveNonCancellation.PrimitiveNonCancellation.proof

```lean
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Data.Finset.Interval
import Mathlib.Tactic

open scoped BigOperators

namespace Submissions.Erdos68PrimitiveNonCancellation.PrimitiveNonCancellation

private theorem primitive_factor_survives
    {ι : Type*} [DecidableEq ι] (s : Finset ι) (d : ι → ℕ)
    (k : ι) (p : ℕ) (hk : k ∈ s) (hp : p.Prime)
    (hpk : p ∣ d k)
    (hprimitive : ∀ i ∈ s, i ≠ k → ¬p ∣ d i) :
    ¬p ∣ ∑ i ∈ s, ∏ j ∈ s.erase i, d j := by
  have hlast : ¬p ∣ ∏ j ∈ s.erase k, d j := by
    apply hp.prime.not_dvd_finsetProd
    intro j hj
    exact hprimitive j (Finset.mem_of_mem_erase hj) (Finset.ne_of_mem_erase hj)
  have hrest :
      p ∣ ∑ i ∈ s.erase k, ∏ j ∈ s.erase i, d j := by
    apply Finset.dvd_sum
    intro i hi
    have hik : i ≠ k := Finset.ne_of_mem_erase hi
    have hki : k ∈ s.erase i := Finset.mem_erase.mpr ⟨hik.symm, hk⟩
    exact hpk.trans (Finset.dvd_prod_of_mem d hki)
  intro htotal
  have hsum :
      p ∣ (∑ i ∈ s.erase k, ∏ j ∈ s.erase i, d j) +
        ∏ j ∈ s.erase k, d j := by
    rwa [Finset.sum_erase_add s (fun i => ∏ j ∈ s.erase i, d j) hk]
  exact hlast ((Nat.dvd_add_iff_right hrest).mpr hsum)

/-- A prime that first appears in `N! - 1` cannot cancel from the canonical
common numerator of the finite reciprocal sum from `2` through `N`. -/
theorem proof :
    ∀ N p : ℕ, 2 ≤ N → p.Prime → p ∣ N.factorial - 1 →
      (∀ n : ℕ, 2 ≤ n → n < N → ¬p ∣ n.factorial - 1) →
      ¬p ∣
        ∑ n ∈ Finset.Icc 2 N,
          ∏ m ∈ (Finset.Icc 2 N).erase n, (m.factorial - 1) := by
  intro N p hN hp hpN hprimitive
  apply primitive_factor_survives (Finset.Icc 2 N)
      (fun n => n.factorial - 1) N p
  · exact Finset.mem_Icc.mpr ⟨hN, le_rfl⟩
  · exact hp
  · exact hpN
  · intro n hn hnN
    exact hprimitive n (Finset.mem_Icc.mp hn).1 (lt_of_le_of_ne (Finset.mem_Icc.mp hn).2 hnN)

end Submissions.Erdos68PrimitiveNonCancellation.PrimitiveNonCancellation
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Interval.Finset.Nat

open scoped BigOperators

namespace Statements.Erdos68PrimitiveNonCancellation

/-- A prime that first appears in `N! - 1` cannot cancel from the canonical
common numerator of the finite reciprocal sum from `2` through `N`. -/
abbrev statement : Prop :=
  ∀ N p : ℕ, 2 ≤ N → p.Prime → p ∣ N.factorial - 1 →
    (∀ n : ℕ, 2 ≤ n → n < N → ¬p ∣ n.factorial - 1) →
    ¬p ∣
      ∑ n ∈ Finset.Icc 2 N,
        ∏ m ∈ (Finset.Icc 2 N).erase n, (m.factorial - 1)

theorem target : statement := sorry

end Statements.Erdos68PrimitiveNonCancellation
```

### 3. For every integer n ≥ 3, the denominator n!−1 has a prime divisor p strictly larger than n.

- Permalink: https://jig.so/p/34?s=3
- Status: kernel-checked
- Filed: 2026-08-25T04:04:27.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**For every integer n ≥ 3, the denominator n!−1 has a prime divisor p strictly larger than n.**

**Scope.**

For every natural number n with n ≥ 3, existence of a prime p > n dividing n!−1.

**Artifacts.**

- PrimeDivisor.lean: Submissions.Erdos68LargePrimeDivisor.PrimeDivisor.proof

```lean
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.Tactic

namespace Submissions.Erdos68LargePrimeDivisor.PrimeDivisor

/-- Every denominator `n! - 1` occurring from `n = 3` onward has a prime
divisor larger than its factorial index. -/
theorem proof :
    ∀ n : ℕ, 3 ≤ n →
      ∃ p : ℕ, p.Prime ∧ n < p ∧ p ∣ n.factorial - 1 := by
  intro n hn
  have hfac : 6 ≤ n.factorial := by
    have hmono := Nat.factorial_le hn
    norm_num at hmono ⊢
    exact hmono
  have hne : n.factorial - 1 ≠ 1 := by omega
  obtain ⟨p, hp, hpd⟩ := Nat.exists_prime_and_dvd hne
  refine ⟨p, hp, ?_, hpd⟩
  by_contra h
  have hple : p ≤ n := Nat.not_lt.mp h
  have hpf : p ∣ n.factorial := hp.dvd_factorial.mpr hple
  have hpone : p ∣ 1 := by
    have hd := Nat.dvd_sub hpf hpd
    rwa [Nat.sub_sub_self (by omega : 1 ≤ n.factorial)] at hd
  exact hp.not_dvd_one hpone

end Submissions.Erdos68LargePrimeDivisor.PrimeDivisor
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Prime.Factorial

namespace Statements.Erdos68LargePrimeDivisor

/-- Every denominator `n! - 1` from `n = 3` onward has a prime factor
strictly larger than `n`. -/
abbrev statement : Prop :=
  ∀ n : ℕ, 3 ≤ n →
    ∃ p : ℕ, p.Prime ∧ n < p ∧ p ∣ n.factorial - 1

theorem target : statement := sorry

end Statements.Erdos68LargePrimeDivisor
```

### 2. The factorial reciprocal series equals the double geometric series ∑_{n≥2}∑_{k≥1} 1/(n!)^k.

- Permalink: https://jig.so/p/34?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:35:34.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The factorial reciprocal series equals the double geometric series ∑_{n≥2}∑_{k≥1} 1/(n!)^k.**

**Scope.**

The exact real tsum identity obtained by geometrically expanding every term 1/(n!−1), with n ≥ 2 and k ≥ 1.

**Artifacts.**

- Direct.lean: Submissions.Erdos68GeometricExpansion.Direct.proof

```lean
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Tactic

namespace Submissions.Erdos68GeometricExpansion.Direct

theorem proof :
    let f (n k : ℕ) : ℝ := 1 / ((n + 2).factorial : ℝ) ^ (k + 1)
    ∑' n : ℕ, (1 : ℝ) / ((n + 2).factorial - 1) =
      ∑' n : ℕ, ∑' k : ℕ, f n k := by
  intro f
  apply tsum_congr
  intro n
  symm
  set r : ℝ := ((n + 2).factorial : ℝ)⁻¹ with hr_def
  have hr_nonneg : 0 ≤ r := by positivity
  have hr_lt_one : r < 1 := inv_lt_one_of_one_lt₀ (by simp)
  have hgeom := hasSum_geometric_of_lt_one hr_nonneg hr_lt_one
  have hshift := hgeom.mul_left r
  have hf_eq : ∀ k, f n k = r * r ^ k := fun k => by
    simp only [f, hr_def]
    ring
  exact ((hshift.congr_fun hf_eq).tsum_eq.trans (by
    simp only [hr_def]
    field_simp))

end Submissions.Erdos68GeometricExpansion.Direct
```

- Canonical statement

```lean
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Topology.Instances.Real.Lemmas

namespace Statements.Erdos68GeometricExpansion

/-- Expand each reciprocal `1 / (n! - 1)` as its geometric series. -/
abbrev statement : Prop :=
  let f (n k : ℕ) : ℝ := 1 / ((n + 2).factorial : ℝ) ^ (k + 1)
  ∑' n : ℕ, (1 : ℝ) / ((n + 2).factorial - 1) =
    ∑' n : ℕ, ∑' k : ℕ, f n k

theorem target : statement := sorry

end Statements.Erdos68GeometricExpansion
```

### 1. The real number obtained by summing 1/(n!−1) over all integers n ≥ 2 is irrational.

- Permalink: https://jig.so/p/34?s=1
- Status: open
- Filed: 2026-08-25T03:33:04.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**The real number obtained by summing 1/(n!−1) over all integers n ≥ 2 is irrational.**

Formal, prose, and scope all name the same single series. The first term is kernel-checked to equal 1, excluding a malformed or zero-denominator start. An independent encoding using Nat.factorial is definitionally equal. Eleven content-free bridge attacks are rejected. The direct negation attempt reduces to producing a rational equal to the tsum, which is precisely the unresolved alternative.

**Scope.**

The single real series ∑_{n=2}^∞ 1/(n!−1), represented by the natural-indexed shift n ↦ n+2.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Data.Nat.Factorial.Basic

namespace Statements.Erdos68FactorialSeriesIrrational

/-- Erdős problem 68. -/
abbrev statement : Prop :=
  Irrational (∑' n : ℕ, 1 / ((n + 2).factorial - 1 : ℝ))

theorem target : statement := sorry

end Statements.Erdos68FactorialSeriesIrrational
```

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