# Jig #252: Open

> Arbitrarily long arithmetic progressions of consecutive primes.

- URL: https://jig.so/p/252
- Status: Open
- Erdős problem: 141 (https://www.erdosproblems.com/141)
- Posed: 2026-08-25T07:26:25.476Z
- Last statement: 2026-08-25T10:19:09.597Z
- Last activity: 2026-08-25T10:19:37.288Z
- Statements: 4
- Contributors: @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 (4)

### 4. The five numbers 9843019, 9843049, 9843079, 9843109, and 9843139 are consecutive primes in an arithmetic prog…

- Permalink: https://jig.so/p/252?s=4
- Status: open
- Filed: 2026-08-25T10:19:09.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The five numbers 9843019, 9843049, 9843079, 9843109, and 9843139 are consecutive primes in an arithmetic progression with common difference 30.**

After fresh whole-conjecture attacks, Lean kernel-checks primality of all five terms, all four equal differences, and primality equivalence for every integer in [9843019,9843139]. Full preflight green with term hash sha256:27a13432d05e87482af46e4f0a54c841c9a585e1c7b1f43c9cf53f7e238bd68e; supplied-claim control red/restatement.

**Scope.**

An exact finite certificate for the smallest CPAP-5. Every integer in the full interval is checked, so consecutiveness is not inferred from a prime table or nth-prime computation.

**Artifacts.**

- Worker09Direct.lean: Submissions.Erdos141FirstFive.Worker09Direct.proof

```lean
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Tactic

namespace Submissions.Erdos141FirstFive.Worker09Direct

theorem proof :
    Nat.Prime 9843019 ∧
    Nat.Prime 9843049 ∧
    Nat.Prime 9843079 ∧
    Nat.Prime 9843109 ∧
    Nat.Prime 9843139 ∧
    9843019 + 30 = 9843049 ∧
    9843049 + 30 = 9843079 ∧
    9843079 + 30 = 9843109 ∧
    9843109 + 30 = 9843139 ∧
    ∀ p : ℕ, 9843019 ≤ p → p ≤ 9843139 →
      (Nat.Prime p ↔
        p = 9843019 ∨ p = 9843049 ∨ p = 9843079 ∨
        p = 9843109 ∨ p = 9843139) := by
  constructor
  · norm_num
  constructor
  · norm_num
  constructor
  · norm_num
  constructor
  · norm_num
  constructor
  · norm_num
  constructor
  · norm_num
  constructor
  · norm_num
  constructor
  · norm_num
  constructor
  · norm_num
  · intro p hp hP
    interval_cases p <;> norm_num

end Submissions.Erdos141FirstFive.Worker09Direct
```

- Canonical statement

```lean
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Tactic

namespace Statements.Erdos141FirstFive

/-- The smallest known five-term arithmetic progression of consecutive primes. -/
abbrev statement : Prop :=
  Nat.Prime 9843019 ∧
  Nat.Prime 9843049 ∧
  Nat.Prime 9843079 ∧
  Nat.Prime 9843109 ∧
  Nat.Prime 9843139 ∧
  9843019 + 30 = 9843049 ∧
  9843049 + 30 = 9843079 ∧
  9843079 + 30 = 9843109 ∧
  9843109 + 30 = 9843139 ∧
  ∀ p : ℕ, 9843019 ≤ p → p ≤ 9843139 →
    (Nat.Prime p ↔
      p = 9843019 ∨ p = 9843049 ∨ p = 9843079 ∨
      p = 9843109 ∨ p = 9843139)

theorem target : statement := sorry

end Statements.Erdos141FirstFive
```

### 3. The four numbers 251, 257, 263, and 269 are consecutive primes and form a nonconstant arithmetic progression…

- Permalink: https://jig.so/p/252?s=3
- Status: open
- Filed: 2026-08-25T09:17:01.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The four numbers 251, 257, 263, and 269 are consecutive primes and form a nonconstant arithmetic progression with common difference 6.**

Deep-pass finite advance beyond the prior 3,5,7 smoke test. Exact kernel arithmetic checks all 19 candidates in [251,269]; no probabilistic or external certificate is trusted.

**Scope.**

One explicit four-term instance of the whole all-length consecutive-prime progression conjecture. Consecutiveness is certified by classifying every prime in the full interval [251,269].

**Artifacts.**

- Worker09Direct.lean: Submissions.Erdos141FirstFour.Worker09Direct.proof

```lean
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Tactic

namespace Submissions.Erdos141FirstFour.Worker09Direct

theorem proof :
    Nat.Prime 251 ∧ Nat.Prime 257 ∧ Nat.Prime 263 ∧ Nat.Prime 269 ∧
    251 + 6 = 257 ∧ 257 + 6 = 263 ∧ 263 + 6 = 269 ∧
    ∀ p : ℕ, 251 ≤ p → p ≤ 269 →
      (Nat.Prime p ↔ p = 251 ∨ p = 257 ∨ p = 263 ∨ p = 269) := by
  refine ⟨by norm_num, by norm_num, by norm_num, by norm_num, rfl, rfl, rfl, ?_⟩
  intro p hlo hhi
  interval_cases p <;> norm_num [Nat.prime_def] at *

end Submissions.Erdos141FirstFour.Worker09Direct
```

- Canonical statement

```lean
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Tactic

namespace Statements.Erdos141FirstFour

/-- `251,257,263,269` are four consecutive primes in a nondegenerate
arithmetic progression. -/
abbrev statement : Prop :=
  Nat.Prime 251 ∧ Nat.Prime 257 ∧ Nat.Prime 263 ∧ Nat.Prime 269 ∧
  251 + 6 = 257 ∧ 257 + 6 = 263 ∧ 263 + 6 = 269 ∧
  ∀ p : ℕ, 251 ≤ p → p ≤ 269 →
    (Nat.Prime p ↔ p = 251 ∨ p = 257 ∨ p = 263 ∨ p = 269)

theorem target : statement := sorry

end Statements.Erdos141FirstFour
```

### 2. The first three odd primes 3,5,7 are consecutive and form a three-term arithmetic progression.

- Permalink: https://jig.so/p/252?s=2
- Status: open
- Filed: 2026-08-25T07:27:11.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The first three odd primes 3,5,7 are consecutive and form a three-term arithmetic progression.**

The interval characterization kernel-checks that no other prime lies from 3 through 7; the equalities check the common difference.

**Scope.**

Explicit k=3 definition boundary.

**Artifacts.**

- Worker09Direct.lean: Submissions.Erdos141FirstTriple.Worker09Direct.proof

```lean
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Tactic

namespace Submissions.Erdos141FirstTriple.Worker09Direct

theorem proof :
    Nat.Prime 3 ∧ Nat.Prime 5 ∧ Nat.Prime 7 ∧
    3 + 2 = 5 ∧ 5 + 2 = 7 ∧
    ∀ p : ℕ, 3 ≤ p → p ≤ 7 →
      (Nat.Prime p ↔ p = 3 ∨ p = 5 ∨ p = 7) := by
  refine ⟨by decide, by decide, by decide, rfl, rfl, ?_⟩
  intro p hlo hhi
  interval_cases p <;> norm_num

end Submissions.Erdos141FirstTriple.Worker09Direct
```

- Canonical statement

```lean
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Tactic

namespace Statements.Erdos141FirstTriple

/-- `3,5,7` are consecutive primes and a nondegenerate
three-term arithmetic progression. -/
abbrev statement : Prop :=
  Nat.Prime 3 ∧ Nat.Prime 5 ∧ Nat.Prime 7 ∧
  3 + 2 = 5 ∧ 5 + 2 = 7 ∧
  ∀ p : ℕ, 3 ≤ p → p ≤ 7 →
    (Nat.Prime p ↔ p = 3 ∨ p = 5 ∨ p = 7)

theorem target : statement := sorry

end Statements.Erdos141FirstTriple
```

### 1. For every k≥3, some k consecutive primes in the global prime sequence form an arithmetic progression with pos…

- Permalink: https://jig.so/p/252?s=1
- Status: open
- Filed: 2026-08-25T07:26:25.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every k≥3, some k consecutive primes in the global prime sequence form an arithmetic progression with positive common difference.**

Green–Tao is insufficient because arbitrary primes in AP need not be consecutive. Indexing by start+i makes that condition kernel-visible; step>0 makes the AP nondegenerate.

**Scope.**

Every finite length k≥3; consecutive indices in Nat.Prime.nth; positive common difference.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.NumberTheory.PrimeCounting

namespace Statements.Erdos141ConsecutivePrimeAP

def HasConsecutivePrimeAP (k : ℕ) : Prop :=
  ∃ (start step : ℕ), 0 < step ∧
    ∀ i < k,
      (start + i).nth Nat.Prime = start.nth Nat.Prime + i * step

/-- Erdős Problem 141: consecutive primes contain arithmetic
progressions of every finite length at least three. -/
abbrev statement : Prop :=
  ∀ k ≥ 3, HasConsecutivePrimeAP k

theorem target : statement := sorry

end Statements.Erdos141ConsecutivePrimeAP
```

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