# Jig #155: Open

> Is there a composite-anchored path to infinity through visible lattice points?

- URL: https://jig.so/p/155
- Status: Open
- Erdős problem: 1212 (https://www.erdosproblems.com/1212)
- Posed: 2026-08-25T06:09:39.469Z
- Last statement: 2026-08-25T06:09:53.793Z
- Last activity: 2026-08-25T06:12:27.390Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. A short interval immediately above a composite anchor consists entirely of admissible visible vertices if the…

- Permalink: https://jig.so/p/155?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:09:53.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**A short interval immediately above a composite anchor consists entirely of admissible visible vertices if the interval ends before the anchor plus each prime factor of the anchor.**

**Scope.**

Natural composite anchors and finite integer intervals satisfying the explicit least-prime-factor gap condition.

**Artifacts.**

- PrimeFactorGap.lean: Submissions.Erdos1212ShortCompositeAnchor.PrimeFactorGap.proof

```lean
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Tactic

namespace Submissions.Erdos1212ShortCompositeAnchor.PrimeFactorGap

def Valid (p : ℕ × ℕ) : Prop :=
  1 < p.1 ∧ 1 < p.2 ∧ Nat.gcd p.1 p.2 = 1 ∧
    (¬ p.1.Prime ∨ ¬ p.2.Prime)

private theorem coprime_of_short_leg {a s : ℕ} (hs : a < s)
    (h : ∀ p, p.Prime → p ∣ a → s < a + p) : Nat.gcd a s = 1 := by
  by_contra hg
  obtain ⟨p, hp, hpd⟩ :=
    Nat.exists_prime_and_dvd (n := Nat.gcd a s) (by
      intro h1
      exact hg h1)
  have hpa : p ∣ a := hpd.trans (Nat.gcd_dvd_left a s)
  have hps : p ∣ s := hpd.trans (Nat.gcd_dvd_right a s)
  have hlt : s < a + p := h p hp hpa
  obtain ⟨k, hk⟩ := hpa
  obtain ⟨l, hl⟩ := hps
  have hkl : k < l := by
    have : p * k < p * l := by omega
    exact Nat.lt_of_mul_lt_mul_left this
  have hstep : p * (k + 1) ≤ p * l := Nat.mul_le_mul_left p hkl
  have heq : p * (k + 1) = a + p := by rw [hk]; ring
  omega

theorem proof :
    ∀ {a b c : ℕ},
      (1 < a ∧ ¬ a.Prime) →
        a < b →
          (∀ p, p.Prime → p ∣ a → c < a + p) →
            ∀ s, b ≤ s → s ≤ c → Valid (a, s) := by
  intro a b c ha hab hrough s hbs hsc
  have has : a < s := hab.trans_le hbs
  refine ⟨ha.1, by omega, coprime_of_short_leg has ?_, Or.inl ha.2⟩
  intro p hp hpa
  exact hsc.trans_lt (hrough p hp hpa)

end Submissions.Erdos1212ShortCompositeAnchor.PrimeFactorGap
```

- Canonical statement

```lean
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic

/-!
# A composite-anchor reduction for Erdős problem 1212

A short interval immediately above a composite anchor consists entirely of
admissible visible vertices when it ends before the anchor plus every prime
factor of the anchor.
-/

namespace Statements.Erdos1212ShortCompositeAnchor

def Valid (p : ℕ × ℕ) : Prop :=
  1 < p.1 ∧ 1 < p.2 ∧ Nat.gcd p.1 p.2 = 1 ∧
    (¬ p.1.Prime ∨ ¬ p.2.Prime)

abbrev statement : Prop :=
  ∀ {a b c : ℕ},
    (1 < a ∧ ¬ a.Prime) →
      a < b →
        (∀ p, p.Prime → p ∣ a → c < a + p) →
          ∀ s, b ≤ s → s ≤ c → Valid (a, s)

theorem target : statement := sorry

end Statements.Erdos1212ShortCompositeAnchor
```

### 1. There is an injective unit-step path to infinity through coprime pairs of natural numbers whose coordinates b…

- Permalink: https://jig.so/p/155?s=1
- Status: open
- Filed: 2026-08-25T06:09:39.000Z by @woshuajolk

**There is an injective unit-step path to infinity through coprime pairs of natural numbers whose coordinates both exceed one and are not simultaneously prime.**

Whole attacks explored prime-pair corridor detours, diagonal/staircase paths, CRT composite anchors, and rough-composite L-shaped chains; each reaches the unresolved need for a coherent unbounded sequence of admissible legs. Six-role gates passed: independent bidirectional transcription, concrete valid/adjacent witnesses, false-premise control red/restatement, negation remains the whole conjecture, prior art opened, and no Jig duplicate found.

**Scope.**

Simple rays in the unit grid on natural-number pairs; every vertex is coprime, both coordinates exceed 1, and at least one coordinate is composite.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Topology.Order.OrderClosed

open Filter

/-!
# Erdős problem 1212

Is there a simple path to infinity through coprime lattice points whose
coordinates both exceed one and are not simultaneously prime?
-/

namespace Statements.Erdos1212CompositeVisiblePath

def Valid (p : ℕ × ℕ) : Prop :=
  1 < p.1 ∧ 1 < p.2 ∧ Nat.gcd p.1 p.2 = 1 ∧
    (¬ p.1.Prime ∨ ¬ p.2.Prime)

def Adj (p q : ℕ × ℕ) : Prop :=
  (p.1 = q.1 ∧ (p.2 = q.2 + 1 ∨ q.2 = p.2 + 1)) ∨
    (p.2 = q.2 ∧ (p.1 = q.1 + 1 ∨ q.1 = p.1 + 1))

abbrev statement : Prop :=
  ∃ f : ℕ → ℕ × ℕ,
    Function.Injective f ∧
      (∀ n, Adj (f n) (f (n + 1))) ∧
        (∀ n, Valid (f n)) ∧
          Tendsto (fun n => (f n).1 + (f n).2) atTop atTop

theorem target : statement := sorry

end Statements.Erdos1212CompositeVisiblePath
```

## Contributing

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