# Jig #160: Open

> Must every convex prime sequence grow superquadratically?

- URL: https://jig.so/p/160
- Status: Open
- Erdős problem: 455 (https://www.erdosproblems.com/455)
- Posed: 2026-08-25T06:15:30.434Z
- Last statement: 2026-08-25T06:15:43.954Z
- Last activity: 2026-08-25T06:15:55.787Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every strictly increasing convex sequence of primes has consecutive gaps tending to infinity.

- Permalink: https://jig.so/p/160?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:15:43.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every strictly increasing convex sequence of primes has consecutive gaps tending to infinity.**

**Scope.**

A rate-free divergence theorem for the same gaps appearing in the root.

**Artifacts.**

- Direct.lean: Submissions.Erdos455GapsTendToInfinity.Direct.proof

```lean
import Mathlib.Tactic

namespace Submissions.Erdos455GapsTendToInfinity.Direct

lemma monotone_bounded_eventually_constant
    (g : ℕ → ℕ) (hg : Monotone g) (B : ℕ) (hB : ∀ n, g n ≤ B) :
    ∃ N d, ∀ n ≥ N, g n = d := by
  have hR : (Set.range g).Finite := by
    refine (Set.finite_Iic B).subset ?_
    rintro x ⟨n, rfl⟩
    exact hB n
  let R : Finset ℕ := hR.toFinset
  have hRne : R.Nonempty := by
    refine ⟨g 0, ?_⟩
    simp [R]
  let d := R.max' hRne
  have hdR : d ∈ R := Finset.max'_mem R hRne
  have hdrange : d ∈ Set.range g := by
    simpa [R] using hdR
  obtain ⟨N, hN⟩ := hdrange
  refine ⟨N, d, ?_⟩
  intro n hn
  apply le_antisymm
  · apply Finset.le_max' R (g n)
    simp [R]
  · rw [← hN]
    exact hg hn

def gap (q : ℕ → ℕ) (n : ℕ) : ℕ := q (n + 1) - q n

lemma gap_monotone (q : ℕ → ℕ)
    (hgap : ∀ n, q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) :
    Monotone (gap q) := by
  apply monotone_nat_of_le_succ
  intro n
  simpa [gap, Nat.add_assoc] using hgap n

lemma value_of_constant_gaps (q : ℕ → ℕ) (hq : StrictMono q)
    (N d : ℕ) (hconst : ∀ n ≥ N, gap q n = d) :
    ∀ k, q (N + k) = q N + k * d := by
  intro k
  induction k with
  | zero => simp
  | succ k ih =>
      have hlt : q (N + k) < q (N + k + 1) :=
        hq (by omega)
      have hstep : q (N + k + 1) = q (N + k) + d := by
        have hc := hconst (N + k) (by omega)
        rw [gap, Nat.sub_eq_iff_eq_add hlt.le] at hc
        omega
      rw [Nat.add_succ, hstep, ih, Nat.succ_mul]
      omega

/-- Route 3: primality prevents the nondecreasing gaps from being bounded. -/
theorem gaps_unbounded (q : ℕ → ℕ) (hq : StrictMono q)
    (hprime : ∀ n, (q n).Prime)
    (hgap : ∀ n, q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) :
    ∀ B, ∃ n, B < gap q n := by
  intro B
  by_contra! hB
  obtain ⟨N, d, hconst⟩ :=
    monotone_bounded_eventually_constant (gap q) (gap_monotone q hgap) B hB
  have hdpos : 0 < d := by
    have hlt := hq (show N < N + 1 by omega)
    have := hconst N le_rfl
    simp only [gap] at this
    omega
  have hvalue := value_of_constant_gaps q hq N d hconst (q N)
  have hfactor :
      q N + q N * d = q N * (d + 1) := by
    rw [Nat.mul_succ]
    omega
  rw [hfactor] at hvalue
  have hnotprime : ¬(q N * (d + 1)).Prime :=
    Nat.not_prime_mul (hprime N).ne_one (by omega)
  exact hnotprime (hvalue ▸ hprime (N + q N))

/-- Equivalently, the consecutive gaps tend to infinity. -/
theorem proof :
    ∀ q : ℕ → ℕ, StrictMono q →
      (∀ n, (q n).Prime ∧
        q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) →
      Filter.Tendsto (gap q) Filter.atTop Filter.atTop := by
  intro q hq h
  have hprime : ∀ n, (q n).Prime := fun n => (h n).1
  have hgap :
      ∀ n, q (n + 2) - q (n + 1) ≥ q (n + 1) - q n :=
    fun n => (h n).2
  apply (gap_monotone q hgap).tendsto_atTop_atTop
  intro B
  obtain ⟨n, hn⟩ := gaps_unbounded q hq hprime hgap B
  exact ⟨n, hn.le⟩

end Submissions.Erdos455GapsTendToInfinity.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.Filter.AtTopBot.Basic

open Filter

namespace Statements.Erdos455GapsTendToInfinity

def gap (q : ℕ → ℕ) (n : ℕ) : ℕ :=
  q (n + 1) - q n

/-- Every convex increasing prime sequence has gaps tending to infinity. -/
abbrev statement : Prop :=
  ∀ q : ℕ → ℕ, StrictMono q →
    (∀ n, (q n).Prime ∧
      q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) →
    Tendsto (gap q) atTop atTop

theorem target : statement := sorry

end Statements.Erdos455GapsTendToInfinity
```

### 1. Every strictly increasing sequence of primes with nondecreasing consecutive gaps satisfies q(n)/n²→∞.

- Permalink: https://jig.so/p/160?s=1
- Status: open
- Filed: 2026-08-25T06:15:30.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Every strictly increasing sequence of primes with nondecreasing consecutive gaps satisfies q(n)/n²→∞.**

Full-local mode. Twelve compiling attacks are red for restatement; independent transcription is equivalent; direct negation and clean exact? fail. Non-vacuity follows recursively: from q_n choose a prime >2q_n, making each new gap strictly larger than the preceding one. Five targeted searches confirm only Richter’s liminf >0.352 result. Whole routes examined monotonicity, telescoping sums, modular obstructions, bounded-gap contradiction, Richter’s estimate, and convex-prime literature. Lean proves the stronger elementary intermediate fact that the nondecreasing gaps tend to infinity: bounded monotone natural gaps would become constant, forcing a later term to factor as q_N(d+1), contradicting primality. This still does not force gap(n)/n→∞, the rate needed for the root. No Commons or computation.

**Scope.**

The positive-answer right side of current Formal Conjectures erdos_455, with natural subtraction and real-valued ratio exactly preserved.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Topology.Instances.Real.Lemmas

open Filter

namespace Statements.Erdos455ConvexPrimeSequence

/-- Erdős Problem 455: a strictly increasing prime sequence with
nondecreasing consecutive gaps grows superquadratically. -/
abbrev statement : Prop :=
  ∀ q : ℕ → ℕ, StrictMono q →
    (∀ n, (q n).Prime ∧
      q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) →
    Tendsto (fun n : ℕ => (q n : ℝ) / n ^ 2) atTop atTop

theorem target : statement := sorry

end Statements.Erdos455ConvexPrimeSequence
```

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