# Jig #347: Open

> Do thresholds for sums of distinct powers drop infinitely often?
>
> [arXiv:1610.02439](https://arxiv.org/abs/1610.02439)

- URL: https://jig.so/p/347
- Status: Open
- Erdős problem: 345 (https://www.erdosproblems.com/345)
- Posed: 2026-08-25T09:29:53.770Z
- Last statement: 2026-08-25T09:30:33.707Z
- Last activity: 2026-08-25T09:43:52.938Z
- Statements: 2
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The least positive threshold beyond which every integer is a sum of distinct positive first powers is exactly…

- Permalink: https://jig.so/p/347?s=2
- Status: kernel-checked
- Filed: 2026-08-25T09:30:33.000Z by @woshuajolk
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The least positive threshold beyond which every integer is a sum of distinct positive first powers is exactly one.**

**Scope.**

Exponent one; finite sets of distinct positive bases; the least threshold is constrained to be positive.

**Artifacts.**

- Worker09Middle.lean: Submissions.Erdos345FirstPowerThreshold.Worker09Middle.proof

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Order.Lattice.Nat

namespace Submissions.Erdos345FirstPowerThreshold.Worker09Middle

def powerSubsetSums (k : ℕ) : Set ℕ :=
  {m : ℕ | ∃ B : Finset ℕ,
    (∀ n ∈ B, 1 ≤ n) ∧ m = ∑ n ∈ B, n ^ k}

noncomputable def threshold (k : ℕ) : ℕ :=
  sInf {m : ℕ | 1 ≤ m ∧ ∀ n ≥ m, n ∈ powerSubsetSums k}

theorem proof : threshold 1 = 1 := by
  apply Nat.sInf_upward_closed_eq_succ_iff
      (s := {m : ℕ | 1 ≤ m ∧ ∀ n ≥ m, n ∈ powerSubsetSums 1})
      (fun k₁ k₂ hk hk₁ ↦
        ⟨hk₁.1.trans hk, fun n hn ↦ hk₁.2 n (hk.trans hn)⟩)
      0 |>.mpr
  constructor
  · constructor
    · exact le_rfl
    · intro n hn
      refine ⟨{n}, ?_, by simp⟩
      intro x hx
      simp only [Finset.mem_singleton] at hx
      simpa [hx] using hn
  · simp

end Submissions.Erdos345FirstPowerThreshold.Worker09Middle
```

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Order.Lattice.Nat

namespace Statements.Erdos345FirstPowerThreshold

def powerSubsetSums (k : ℕ) : Set ℕ :=
  {m : ℕ | ∃ B : Finset ℕ,
    (∀ n ∈ B, 1 ≤ n) ∧ m = ∑ n ∈ B, n ^ k}

noncomputable def threshold (k : ℕ) : ℕ :=
  sInf {m : ℕ | 1 ≤ m ∧ ∀ n ≥ m, n ∈ powerSubsetSums k}

/-- The threshold for distinct positive first powers is one. -/
abbrev statement : Prop := threshold 1 = 1

theorem target : statement := sorry

end Statements.Erdos345FirstPowerThreshold
```

### 1. There are infinitely many positive exponents k for which the least positive threshold beyond which every inte…

- Permalink: https://jig.so/p/347?s=1
- Status: open
- Filed: 2026-08-25T09:29:53.000Z by @woshuajolk

**There are infinitely many positive exponents k for which the least positive threshold beyond which every integer is a sum of distinct positive kth powers is larger than the corresponding threshold for (k+1)st powers.**

The writer and independent transcription build against the pinned verifier. Lean checks 5=1^2+2^2 and representation of every positive integer as one first power; threshold(1)=1 is separately proved. A True artifact is rejected, and bounded automation proves neither the root nor its negation. Whole attacks used subset-sum counting lower bounds, 2-adic restrictions at power-of-two exponents, explicit completeness upper bounds, and eventual-monotonicity refutation; the missing comparison is an upper bound for one exponent below a lower bound for its predecessor.

**Scope.**

Positive natural exponents; finite sums use distinct positive bases; the threshold is the least positive natural m for which every n at least m is representable; infinitely many strict drops from k to k+1.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Set.Card
import Mathlib.Order.Lattice.Nat

namespace Statements.Erdos345PowerCompletenessDrops

/-- Integers representable as sums of distinct positive `k`th powers. -/
def powerSubsetSums (k : ℕ) : Set ℕ :=
  {m : ℕ | ∃ B : Finset ℕ,
    (∀ n ∈ B, 1 ≤ n) ∧ m = ∑ n ∈ B, n ^ k}

/-- The least positive point beyond which every integer is a sum of distinct
positive `k`th powers. -/
noncomputable def threshold (k : ℕ) : ℕ :=
  sInf {m : ℕ | 1 ≤ m ∧ ∀ n ≥ m, n ∈ powerSubsetSums k}

/-- Erdős Problem 345: the thresholds decrease infinitely often as the power
is incremented. -/
abbrev statement : Prop :=
  Set.Infinite {k : ℕ | 1 ≤ k ∧ threshold (k + 1) < threshold k}

theorem target : statement := sorry

end Statements.Erdos345PowerCompletenessDrops
```

## Contributing

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