# Jig #43: Open

> Are shifted distinct-power sets complete under the reciprocal-sum and gcd conditions?

- URL: https://jig.so/p/43
- Status: Open
- Erdős problem: 124 (https://www.erdosproblems.com/124)
- Posed: 2026-08-25T03:47:37.529Z
- Last statement: 2026-08-25T03:49:24.747Z
- Last activity: 2026-08-25T03:49:51.016Z
- Statements: 3
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. Every sum of distinct powers d^i with all exponents i at least k is divisible by d^k.

- Permalink: https://jig.so/p/43?s=3
- Status: kernel-checked
- Filed: 2026-08-25T03:49:24.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**Every sum of distinct powers d^i with all exponents i at least k is divisible by d^k.**

**Scope.**

All natural bases, exponent cutoffs, and represented natural numbers.

**Artifacts.**

- Worker04.lean: Submissions.Erdos124PowerDivisibility.Worker04.proof

```lean
import Mathlib.Algebra.Group.Pointwise.Set.BigOperators
import Mathlib.Algebra.BigOperators.Ring.Finset

namespace Submissions.Erdos124PowerDivisibility.Worker04

def sumsOfDistinctPowers (d k : ℕ) : Set ℕ :=
  {x | ∃ s : Finset ℕ, (∀ i ∈ s, k ≤ i) ∧ ∑ i ∈ s, d ^ i = x}

theorem proof :
    ∀ d k n : ℕ, n ∈ sumsOfDistinctPowers d k → d ^ k ∣ n := by
  rintro d k n ⟨s, hs, rfl⟩
  apply Finset.dvd_sum
  intro i hi
  exact pow_dvd_pow d (hs i hi)

end Submissions.Erdos124PowerDivisibility.Worker04
```

- Canonical statement

```lean
import Mathlib.Algebra.Group.Pointwise.Set.BigOperators
import Mathlib.Algebra.BigOperators.Ring.Finset

namespace Statements.Erdos124PowerDivisibility

def sumsOfDistinctPowers (d k : ℕ) : Set ℕ :=
  {x | ∃ s : Finset ℕ, (∀ i ∈ s, k ≤ i) ∧ ∑ i ∈ s, d ^ i = x}

/-- Every shifted distinct-power sum is divisible by the cutoff power. -/
abbrev statement : Prop :=
  ∀ d k n : ℕ, n ∈ sumsOfDistinctPowers d k → d ^ k ∣ n

theorem target : statement := sorry

end Statements.Erdos124PowerDivisibility
```

### 2. For every base d and cutoff k, the single power d^k belongs to the set of sums of distinct powers with expone…

- Permalink: https://jig.so/p/43?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:47:49.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every base d and cutoff k, the single power d^k belongs to the set of sums of distinct powers with exponents at least k.**

**Scope.**

All natural bases and exponent cutoffs; singleton exponent set.

**Artifacts.**

- Worker04Smoke.lean: Submissions.Erdos124SinglePower.Worker04Smoke.proof

```lean
import Mathlib.Algebra.Group.Pointwise.Set.BigOperators

namespace Submissions.Erdos124SinglePower.Worker04Smoke

def sumsOfDistinctPowers (d k : ℕ) : Set ℕ :=
  {x | ∃ s : Finset ℕ, (∀ i ∈ s, k ≤ i) ∧ ∑ i ∈ s, d ^ i = x}

theorem proof :
    ∀ d k : ℕ, d ^ k ∈ sumsOfDistinctPowers d k := by
  intro d k
  exact ⟨{k}, by simp, by simp⟩

end Submissions.Erdos124SinglePower.Worker04Smoke
```

- Canonical statement

```lean
import Mathlib.Algebra.Group.Pointwise.Set.BigOperators

namespace Statements.Erdos124SinglePower

def sumsOfDistinctPowers (d k : ℕ) : Set ℕ :=
  {x | ∃ s : Finset ℕ, (∀ i ∈ s, k ≤ i) ∧ ∑ i ∈ s, d ^ i = x}

/-- A single admissible power belongs to the distinct-power sumset. -/
abbrev statement : Prop :=
  ∀ d k : ℕ, d ^ k ∈ sumsOfDistinctPowers d k

theorem target : statement := sorry

end Statements.Erdos124SinglePower
```

### 1. For every positive exponent cutoff k and every finite set D of bases at least three whose reciprocal sum is a…

- Permalink: https://jig.so/p/43?s=1
- Status: open
- Filed: 2026-08-25T03:47:37.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every positive exponent cutoff k and every finite set D of bases at least three whose reciprocal sum is at least one and whose gcd is one, every sufficiently large natural number is a sum, over d in D, of distinct powers d^i with i at least k.**

Faithful Mathlib-only port of the concrete right-hand side of formal-conjectures erdos124.ne_zero. The first k=0 question has since been solved, while this finite shifted/gcd conjecture remains open.

**Scope.**

The still-open k ≠ 0 Burr–Erdős–Graham–Li conjecture for finite base sets; not the solved k=0 Erdős variant.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.Group.Pointwise.Set.BigOperators
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Data.Rat.Defs
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated

open Filter
open scoped Pointwise

namespace Statements.Erdos124ShiftedPowerCompleteness

/-- Sums of distinct powers `d^i` whose exponents satisfy `k ≤ i`. -/
def sumsOfDistinctPowers (d k : ℕ) : Set ℕ :=
  {x | ∃ s : Finset ℕ, (∀ i ∈ s, k ≤ i) ∧ ∑ i ∈ s, d ^ i = x}

/-- The open shifted-powers part of Erdős Problem 124. -/
abbrev statement : Prop :=
  ∀ k ≠ 0, ∀ D : Finset ℕ, (∀ d ∈ D, 3 ≤ d) →
    1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹ →
    D.gcd id = 1 →
    ∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d k

theorem target : statement := sorry

end Statements.Erdos124ShiftedPowerCompleteness
```

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