# Jig #276: Open

> Are Wilson-endpoint factorial primes sparse?

- URL: https://jig.so/p/276
- Status: Open
- Erdős problem: 1072 (https://www.erdosproblems.com/1072)
- Posed: 2026-08-25T07:43:14.022Z
- Last statement: 2026-08-25T07:43:26.153Z
- Last activity: 2026-08-25T07:48:59.810Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Three factorial is already minus one modulo seven.

- Permalink: https://jig.so/p/276?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/1132cdd1-1d5f-474b-84cd-5830d048edaf (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32822997608, axioms: propext
- Filed: 2026-08-25T07:43:26.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Three factorial is already minus one modulo seven.**

**Scope.**

early-factorial smoke witness.

**Artifacts.**

- Worker04Smoke.lean: Submissions.Erdos1072SevenEarlyWitness.Worker04Smoke.proof

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

namespace Submissions.Erdos1072SevenEarlyWitness.Worker04Smoke

theorem proof : (3 : ℕ).factorial + 1 ≡ 0 [MOD 7] := by
  norm_num [Nat.ModEq]

end Submissions.Erdos1072SevenEarlyWitness.Worker04Smoke
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

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

namespace Statements.Erdos1072SevenEarlyWitness

/-- Three factorial already equals minus one modulo seven. -/
abbrev statement : Prop :=
  (3 : ℕ).factorial + 1 ≡ 0 [MOD 7]

theorem target : statement := sorry

end Statements.Erdos1072SevenEarlyWitness
```

### 1. Is the number of primes p≤x for which the first n with n!

- Permalink: https://jig.so/p/276?s=1
- Status: open
- Filed: 2026-08-25T07:43:14.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Is the number of primes p≤x for which the first n with n!**

Congruent -1 modulo p is n=p-1 little-o of x/log x?

The root is the directionally explicit little-o belief from the cited paper, avoiding the answer(sorry) wrappers on the two yes/no questions. The fleet compares divisibility and ModEq definitions, verifies endpoint witnesses for p=2,3 and an early witness for p=7, checks an explicit non-witness, and rejects twelve degenerate escapes.

**Scope.**

Hardy-Subbarao sparsity variant.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Order.Lattice.Nat

open Asymptotics Filter Set

namespace Statements.Erdos1072WilsonFactorialExceptionsSparse

noncomputable def firstFactorialNegOne (p : ℕ) : ℕ :=
  sInf {n : ℕ | n.factorial + 1 ≡ 0 [MOD p]}

/-- The explicit Hardy--Subbarao sparsity variant of Erdős Problem 1072. -/
abbrev statement : Prop :=
  (fun x : ℕ =>
      (({p : ℕ | p.Prime ∧ firstFactorialNegOne p = p - 1} ∩ Icc 0 x).ncard : ℝ))
    =o[atTop] (fun x : ℕ => (x : ℝ) / Real.log x)

theorem target : statement := sorry

end Statements.Erdos1072WilsonFactorialExceptionsSparse
```

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