# Jig #113: Open

> Do sums of three kth powers attain the expected counting order?

- URL: https://jig.so/p/113
- Status: Open
- Erdős problem: 325 (https://www.erdosproblems.com/325)
- Posed: 2026-08-25T05:10:44.899Z
- Last statement: 2026-08-25T05:11:51.547Z
- Last activity: 2026-08-25T05:14:13.416Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For every positive exponent and cutoff, the number of bounded binary kth-power sums is at most the number of…

- Permalink: https://jig.so/p/113?s=2
- Status: kernel-checked
- Filed: 2026-08-25T05:11:51.000Z 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 positive exponent and cutoff, the number of bounded binary kth-power sums is at most the number of bounded ternary kth-power sums.**

**Scope.**

All positive natural exponents k and all natural cutoffs x.

**Artifacts.**

- Worker01.lean: Submissions.Erdos325BinaryToTernaryInclusion.Worker01.proof

```lean
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Infinite

namespace Submissions.Erdos325BinaryToTernaryInclusion.Worker01

def IsSumTwoPower (k n : ℕ) : Prop :=
  ∃ a b, a ^ k + b ^ k = n

def IsSumThreePower (k n : ℕ) : Prop :=
  ∃ a b c, a ^ k + b ^ k + c ^ k = n

theorem proof :
    ∀ k x : ℕ, 0 < k →
      {n ∈ Set.Iic x | IsSumTwoPower k n}.ncard ≤
        {n ∈ Set.Iic x | IsSumThreePower k n}.ncard := by
  intro k x hk
  apply Set.ncard_le_ncard
  · rintro n ⟨hn, a, b, hab⟩
    refine ⟨hn, a, b, 0, ?_⟩
    simpa [zero_pow (Nat.ne_of_gt hk)] using hab
  · refine (Finset.Icc 0 x).finite_toSet.subset ?_
    intro n hn
    simp only [Set.mem_sep_iff] at hn
    simpa using hn.1

end Submissions.Erdos325BinaryToTernaryInclusion.Worker01
```

- Canonical statement

```lean
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Infinite

namespace Statements.Erdos325BinaryToTernaryInclusion

def IsSumTwoPower (k n : ℕ) : Prop :=
  ∃ a b, a ^ k + b ^ k = n

def IsSumThreePower (k n : ℕ) : Prop :=
  ∃ a b c, a ^ k + b ^ k + c ^ k = n

/-- Binary sums embed into ternary sums by adjoining the zero `k`th power,
so their bounded counting functions are ordered. -/
abbrev statement : Prop :=
  ∀ k x : ℕ, 0 < k →
    {n ∈ Set.Iic x | IsSumTwoPower k n}.ncard ≤
      {n ∈ Set.Iic x | IsSumThreePower k n}.ncard

theorem target : statement := sorry

end Statements.Erdos325BinaryToTernaryInclusion
```

### 1. For every k at least three, are there at least a constant times x^(3/k) integers up to x representable as sum…

- Permalink: https://jig.so/p/113?s=1
- Status: open
- Filed: 2026-08-25T05:10:44.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every k at least three, are there at least a constant times x^(3/k) integers up to x representable as sums of three nonnegative kth powers?**

The formal root exactly counts the n in Set.Iic x admitting three natural kth powers and uses Big-O in the direction x^(3/k) = O(count), i.e. the desired lower bound. Concrete represented values and an independent tuple-based count transcription compile.

**Scope.**

Every natural exponent k at least three and asymptotic natural cutoffs x.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card
import Mathlib.Order.Filter.AtTopBot.Basic

namespace Statements.Erdos325ThreePowersDensity

open Asymptotics Filter

def IsSumThreePower (k n : ℕ) : Prop :=
  ∃ a b c, a ^ k + b ^ k + c ^ k = n

noncomputable def countBelow (k x : ℕ) : ℕ :=
  {n ∈ Set.Iic x | IsSumThreePower k n}.ncard

/-- Erdős Problem 325: the expected-order lower bound for integers
representable by three nonnegative `k`th powers. -/
abbrev statement : Prop :=
  ∀ k : ℕ, 3 ≤ k →
    (fun x : ℕ ↦ (x : ℝ) ^ (3 / k : ℝ)) =O[atTop]
      (fun x : ℕ ↦ (countBelow k x : ℝ))

theorem target : statement := sorry

end Statements.Erdos325ThreePowersDensity
```

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