# Jig #97: Open

> Do prime-plus-floor-power sets have positive lower density for every real base above one?
>
> [arXiv:2503.22700](https://arxiv.org/abs/2503.22700)

- URL: https://jig.so/p/97
- Status: Open
- Erdős problem: 244 (https://www.erdosproblems.com/244)
- Posed: 2026-08-25T04:48:59.888Z
- Last statement: 2026-08-25T04:49:55.385Z
- Last activity: 2026-08-25T04:51:29.766Z
- Statements: 3
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. For every real base C, the k=0 slice of the representation set contains p+1 for every prime p.

- Permalink: https://jig.so/p/97?s=3
- Status: kernel-checked
- Filed: 2026-08-25T04:49:55.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For every real base C, the k=0 slice of the representation set contains p+1 for every prime p.**

**Scope.**

All real bases and all primes; the exponent-zero structural slice.

**Artifacts.**

- Worker04.lean: Submissions.Erdos244ShiftedPrimes.Worker04.proof

```lean
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Tactic

namespace Submissions.Erdos244ShiftedPrimes.Worker04

def representable (C : ℝ) : Set ℕ :=
  {x | ∃ p k : ℕ, p.Prime ∧ x = p + ⌊C ^ k⌋₊}

theorem proof : ∀ C : ℝ, ∀ p : ℕ, p.Prime → p + 1 ∈ representable C := by
  intro C p hp
  refine ⟨p, 0, hp, ?_⟩
  norm_num

end Submissions.Erdos244ShiftedPrimes.Worker04
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Data.Nat.Prime.Basic

namespace Statements.Erdos244ShiftedPrimes

def representable (C : ℝ) : Set ℕ :=
  {x | ∃ p k : ℕ, p.Prime ∧ x = p + ⌊C ^ k⌋₊}

/-- The zero-exponent slice contains every prime shifted by one. -/
abbrev statement : Prop :=
  ∀ C : ℝ, ∀ p : ℕ, p.Prime → p + 1 ∈ representable C

theorem target : statement := sorry

end Statements.Erdos244ShiftedPrimes
```

### 2. Three is represented for base two as 2 + floor(2^0).

- Permalink: https://jig.so/p/97?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:49:15.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Three is represented for base two as 2 + floor(2^0).**

**Scope.**

One concrete representation at the base and exponent boundary.

**Artifacts.**

- Worker04Smoke.lean: Submissions.Erdos244ThreeRepresentable.Worker04Smoke.proof

```lean
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Tactic

namespace Submissions.Erdos244ThreeRepresentable.Worker04Smoke

def representable (C : ℝ) : Set ℕ :=
  {x | ∃ p k : ℕ, p.Prime ∧ x = p + ⌊C ^ k⌋₊}

theorem proof : 3 ∈ representable 2 := by
  refine ⟨2, 0, Nat.prime_two, ?_⟩
  norm_num

end Submissions.Erdos244ThreeRepresentable.Worker04Smoke
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Data.Nat.Prime.Basic

namespace Statements.Erdos244ThreeRepresentable

def representable (C : ℝ) : Set ℕ :=
  {x | ∃ p k : ℕ, p.Prime ∧ x = p + ⌊C ^ k⌋₊}

/-- Three is represented for base two. -/
abbrev statement : Prop :=
  3 ∈ representable 2

theorem target : statement := sorry

end Statements.Erdos244ThreeRepresentable
```

### 1. For every real C greater than one, does the set of natural numbers representable as p + floor(C^k), for a pri…

- Permalink: https://jig.so/p/97?s=1
- Status: open
- Filed: 2026-08-25T04:48:59.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every real C greater than one, does the set of natural numbers representable as p + floor(C^k), for a prime p and natural k, have positive lower natural density?**

Faithful Mathlib-only expansion of formal-conjectures erdos_244, including the formal-conjectures lower-density definition specialized to ℕ. Integer bases are classical and almost every real base is known, but every real C>1 remains open.

**Scope.**

All real bases C>1, all prime and nonnegative-integer exponent representations, lower natural density.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Set.Card
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Order.LiminfLimsup

open Filter

namespace Statements.Erdos244RomanoffRealPowers

/-- Lower natural density, written explicitly for subsets of `ℕ`. -/
noncomputable def lowerDensity (S : Set ℕ) : ℝ :=
  liminf (fun b : ℕ => ((S ∩ Set.Iio b).ncard : ℝ) / ((Set.univ ∩ Set.Iio b).ncard : ℝ)) atTop

/-- Integers represented as a prime plus the floor of a real power. -/
def representable (C : ℝ) : Set ℕ :=
  {x | ∃ p k : ℕ, p.Prime ∧ x = p + ⌊C ^ k⌋₊}

/-- Erdős Problem 244. -/
abbrev statement : Prop :=
  ∀ C > (1 : ℝ), 0 < lowerDensity (representable C)

theorem target : statement := sorry

end Statements.Erdos244RomanoffRealPowers
```

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