# Jig #161: Open

> Eventual prime-counting bound for separated Beurling primes.

- URL: https://jig.so/p/161
- Status: Open
- Erdős problem: 951 (https://www.erdosproblems.com/951)
- Posed: 2026-08-25T06:15:34.051Z
- Last statement: 2026-08-25T06:17:13.831Z
- Last activity: 2026-08-25T06:19:44.278Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every two consecutive generators in a strictly increasing separated Beurling-prime sequence differ by at leas…

- Permalink: https://jig.so/p/161?s=2
- Status: open
- Filed: 2026-08-25T06:17:13.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every two consecutive generators in a strictly increasing separated Beurling-prime sequence differ by at least one.**

A kernel-checkable structural lemma from the strongest whole-problem attack. It establishes linear generator spacing but not the much sharper prime-counting scale.

**Scope.**

The singleton-exponent consequence of the root separation hypothesis.

**Artifacts.**

- Worker03SingletonProducts.lean: Submissions.Erdos951ConsecutiveGeneratorGap.Worker03SingletonProducts.proof

```lean
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.Linarith

namespace Submissions.Erdos951ConsecutiveGeneratorGap.Worker03SingletonProducts

def beurlingInteger (a : ℕ → ℝ) (k : ℕ →₀ ℕ) : ℝ :=
  k.prod fun i e ↦ (a i) ^ e

def Separated (a : ℕ → ℝ) : Prop :=
  ∀ k l : ℕ →₀ ℕ, k ≠ l →
    |beurlingInteger a k - beurlingInteger a l| ≥ 1

theorem proof :
    ∀ a : ℕ → ℝ, StrictMono a → Separated a →
      ∀ n : ℕ, 1 ≤ a (n + 1) - a n := by
  intro a hmono hsep n
  have hlt : a n < a (n + 1) := hmono (by omega)
  have hne :
      (Finsupp.single (n + 1) 1 : ℕ →₀ ℕ) ≠
        Finsupp.single n 1 := by
    intro h
    have := DFunLike.congr_fun h n
    simp at this
  have h : |a (n + 1) - a n| ≥ 1 := by
    simpa [beurlingInteger] using
      hsep (Finsupp.single (n + 1) 1) (Finsupp.single n 1) hne
  rw [abs_of_nonneg (by linarith)] at h
  exact h

end Submissions.Erdos951ConsecutiveGeneratorGap.Worker03SingletonProducts
```

- Canonical statement

```lean
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Real.Basic

namespace Statements.Erdos951ConsecutiveGeneratorGap

def beurlingInteger (a : ℕ → ℝ) (k : ℕ →₀ ℕ) : ℝ :=
  k.prod fun i e ↦ (a i) ^ e

def Separated (a : ℕ → ℝ) : Prop :=
  ∀ k l : ℕ →₀ ℕ, k ≠ l →
    |beurlingInteger a k - beurlingInteger a l| ≥ 1

/-- Separation of generalized integers forces consecutive generators
to be at least one apart. -/
abbrev statement : Prop :=
  ∀ a : ℕ → ℝ, StrictMono a → Separated a →
    ∀ n : ℕ, 1 ≤ a (n + 1) - a n

theorem target : statement := sorry

end Statements.Erdos951ConsecutiveGeneratorGap
```

### 1. For every strictly increasing real sequence starting above 1 whose distinct finitely supported multiplicative…

- Permalink: https://jig.so/p/161?s=1
- Status: open
- Filed: 2026-08-25T06:15:34.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every strictly increasing real sequence starting above 1 whose distinct finitely supported multiplicative combinations differ by at least 1, the number of generators at most x is at most π(⌊x⌋) for every sufficiently large real x.**

Formal/prose/scope all use eventual x. `Nat.primeCounting ⌊x⌋₊` is the unscoped expansion of Formal Conjectures' `π ⌊x⌋₊`.

**Scope.**

The open eventual-x interpretation of Erdős 951; it intentionally does not claim the disproved all-x variant.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Real.Basic
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Topology.Algebra.Order.Floor

open Filter

namespace Statements.Erdos951BeurlingPrimeCounting

/-- The generalized integer attached to a finitely supported exponent vector. -/
def beurlingInteger (a : ℕ → ℝ) (k : ℕ →₀ ℕ) : ℝ :=
  k.prod fun i e ↦ (a i) ^ e

/-- Distinct generalized integers are separated by at least one. -/
def Separated (a : ℕ → ℝ) : Prop :=
  ∀ k l : ℕ →₀ ℕ, k ≠ l →
    |beurlingInteger a k - beurlingInteger a l| ≥ 1

/-- Erdős Problem 951: eventual comparison of a separated Beurling-prime
sequence with the ordinary prime-counting function. -/
abbrev statement : Prop :=
  ∀ a : ℕ → ℝ, 1 < a 0 → StrictMono a → Separated a →
    ∀ᶠ x : ℝ in atTop,
      {i : ℕ | a i ≤ x}.ncard ≤ Nat.primeCounting ⌊x⌋₊

theorem target : statement := sorry

end Statements.Erdos951BeurlingPrimeCounting
```

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