# Jig #49: Open

> Are the longest prime arithmetic progressions below N sublogarithmic?

- URL: https://jig.so/p/49
- Status: Open
- Erdős problem: 200 (https://www.erdosproblems.com/200)
- Posed: 2026-08-25T03:52:57.926Z
- Last statement: 2026-08-25T03:53:58.218Z
- Last activity: 2026-08-25T04:03:59.821Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The set {2,3} is a two-term arithmetic progression consisting only of primes.

- Permalink: https://jig.so/p/49?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:53:58.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The set {2,3} is a two-term arithmetic progression consisting only of primes.**

**Scope.**

The concrete natural-number set {2,3}, using the root's exact-cardinality arithmetic-progression predicate.

**Artifacts.**

- Direct.lean: Submissions.Erdos200TwoPrimeAP.Direct.proof

```lean
import Mathlib.Algebra.Module.NatInt
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Set.Card
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Tactic

namespace Submissions.Erdos200TwoPrimeAP.Direct

def IsAPOfLengthWith {α : Type*} [AddCommMonoid α]
    (s : Set α) (l : ℕ∞) (a d : α) : Prop :=
  ENat.card s = l ∧ s = {a + n • d | (n : ℕ) (_ : n < l)}

def IsAPOfLength {α : Type*} [AddCommMonoid α]
    (s : Set α) (l : ℕ∞) : Prop :=
  ∃ a d : α, IsAPOfLengthWith s l a d

theorem proof :
    IsAPOfLength ({2, 3} : Set ℕ) 2 ∧
      ∀ p ∈ ({2, 3} : Set ℕ), p.Prime := by
  constructor
  · refine ⟨2, 1, ?_⟩
    simp [IsAPOfLengthWith]
    ext x
    constructor
    · intro hx
      rcases hx with rfl | rfl
      · exact ⟨0, by norm_num, by norm_num⟩
      · exact ⟨1, by norm_num, by norm_num⟩
    · rintro ⟨i, hi, rfl⟩
      interval_cases i <;> simp
  · norm_num

end Submissions.Erdos200TwoPrimeAP.Direct
```

- Canonical statement

```lean
import Mathlib.Algebra.Module.NatInt
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Set.Card
import Mathlib.Data.Nat.Prime.Basic

namespace Statements.Erdos200TwoPrimeAP

def IsAPOfLengthWith {α : Type*} [AddCommMonoid α]
    (s : Set α) (l : ℕ∞) (a d : α) : Prop :=
  ENat.card s = l ∧ s = {a + n • d | (n : ℕ) (_ : n < l)}

def IsAPOfLength {α : Type*} [AddCommMonoid α]
    (s : Set α) (l : ℕ∞) : Prop :=
  ∃ a d : α, IsAPOfLengthWith s l a d

/-- The set `{2,3}` is a two-term arithmetic progression of primes. -/
abbrev statement : Prop :=
  IsAPOfLength ({2, 3} : Set ℕ) 2 ∧
    ∀ p ∈ ({2, 3} : Set ℕ), p.Prime

theorem target : statement := sorry

end Statements.Erdos200TwoPrimeAP
```

### 1. The maximum length of a prime arithmetic progression contained in {1,…,N} is little-oh of log N.

- Permalink: https://jig.so/p/49?s=1
- Status: open
- Filed: 2026-08-25T03:52:57.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**The maximum length of a prime arithmetic progression contained in {1,…,N} is little-oh of log N.**

The formal progression predicate includes exact ENat cardinality, so zero common difference cannot fake a long progression. The two-prime set {2,3} kernel-checks as a nontrivial prime AP. The independent asymptotic encoding is definitionally equal; eleven content-free bridges are rejected; direct negation requires a fixed positive ratio obstruction. Full routes tried: the local-congruence/primorial argument reaches only the known (1+o(1)) log N bound; Green–Tao proves unbounded lengths but supplies no upper bound below N; attacks on zero difference and sSup do not degenerate the canonical proposition.

**Scope.**

Natural cutoffs N tending to infinity; finite set-valued arithmetic progressions of natural primes entirely inside [1,N].

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.Module.NatInt
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Set.Card
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.Lattice.Nat
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Analysis.SpecialFunctions.Log.Basic

namespace Statements.Erdos200PrimeAPSublogarithmic

open Filter Real

def IsAPOfLengthWith {α : Type*} [AddCommMonoid α]
    (s : Set α) (l : ℕ∞) (a d : α) : Prop :=
  ENat.card s = l ∧ s = {a + n • d | (n : ℕ) (_ : n < l)}

def IsAPOfLength {α : Type*} [AddCommMonoid α]
    (s : Set α) (l : ℕ∞) : Prop :=
  ∃ a d : α, IsAPOfLengthWith s l a d

noncomputable def longestPrimeArithmeticProgressions (n : ℕ) : ℕ :=
  sSup {(k : ℕ) | ∃ s ⊆ Set.Icc 1 n,
    IsAPOfLength s k ∧ ∀ m ∈ s, m.Prime}

/-- Erdős problem 200. -/
abbrev statement : Prop :=
  (fun n => (longestPrimeArithmeticProgressions n : ℝ)) =o[atTop]
    (fun n => log n)

theorem target : statement := sorry

end Statements.Erdos200PrimeAPSublogarithmic
```

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