# Jig #46: Open

> Do infinite Sidon sets attain every exponent below one half?
>
> [arXiv:1209.0326](https://arxiv.org/abs/1209.0326)

- URL: https://jig.so/p/46
- Status: Open
- Erdős problem: 39 (https://www.erdosproblems.com/39)
- Posed: 2026-08-25T03:50:40.715Z
- Last statement: 2026-08-25T03:55:36.081Z
- Last activity: 2026-08-25T03:56:13.418Z
- Statements: 3
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. The powers of three form an infinite Sidon set: every equality between two sums of powers of three agrees ter…

- Permalink: https://jig.so/p/46?s=3
- Status: kernel-checked
- Filed: 2026-08-25T03:55:36.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**The powers of three form an infinite Sidon set: every equality between two sums of powers of three agrees termwise up to swapping.**

**Scope.**

The explicit set {3^n | n ∈ ℕ}, with all unordered two-term sums.

**Artifacts.**

- Worker03Powers3.lean: Submissions.Erdos39InfiniteSidonPowers3.Worker03Powers3.proof

```lean
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Set.Card
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring

namespace Submissions.Erdos39InfiniteSidonPowers3.Worker03Powers3

lemma three_not_dvd_one_add_pow (n : ℕ) : ¬ 3 ∣ 1 + 3 ^ n := by
  cases n with
  | zero => norm_num
  | succ n => simp [pow_succ, Nat.dvd_iff_mod_eq_zero]

lemma factorization_three_add_pow {i j : ℕ} (hij : i ≤ j) :
    (3 ^ i + 3 ^ j).factorization 3 = i := by
  have hfac : 3 ^ i + 3 ^ j = 3 ^ i * (1 + 3 ^ (j - i)) := by
    rw [← Nat.pow_sub_mul_pow 3 hij]
    ring
  rw [hfac, Nat.factorization_mul (by positivity) (by positivity),
    Nat.Prime.factorization_pow (by decide), Finsupp.add_apply,
    Finsupp.single_eq_same,
    Nat.factorization_eq_zero_of_not_dvd (three_not_dvd_one_add_pow (j - i))]
  simp

lemma pow_three_add_pow_three_unique (a b c d : ℕ)
    (h : 3 ^ a + 3 ^ c = 3 ^ b + 3 ^ d) :
    (a = b ∧ c = d) ∨ (a = d ∧ c = b) := by
  have hinj : Function.Injective (fun n : ℕ ↦ 3 ^ n) :=
    Nat.pow_right_injective (by decide)
  by_cases hac : a ≤ c
  · by_cases hbd : b ≤ d
    · have hab : a = b := by
        rw [← factorization_three_add_pow hac, h, factorization_three_add_pow hbd]
      left
      refine ⟨hab, hinj ?_⟩
      exact Nat.add_left_cancel (hab ▸ h)
    · have hdb : d ≤ b := Nat.le_of_lt (Nat.lt_of_not_ge hbd)
      have had : a = d := by
        rw [← factorization_three_add_pow hac, h, add_comm,
          factorization_three_add_pow hdb]
      right
      refine ⟨had, hinj ?_⟩
      exact Nat.add_left_cancel (had ▸ h.trans (add_comm _ _))
  · have hca : c ≤ a := Nat.le_of_lt (Nat.lt_of_not_ge hac)
    by_cases hbd : b ≤ d
    · have hcb : c = b := by
        rw [← factorization_three_add_pow hca, add_comm, h,
          factorization_three_add_pow hbd]
      right
      refine ⟨hinj ?_, hcb⟩
      apply Nat.add_left_cancel
      calc
        3 ^ b + 3 ^ a = 3 ^ a + 3 ^ b := add_comm _ _
        _ = 3 ^ b + 3 ^ d := hcb ▸ h
    · have hdb : d ≤ b := Nat.le_of_lt (Nat.lt_of_not_ge hbd)
      have hcd : c = d := by
        rw [← factorization_three_add_pow hca, add_comm, h, add_comm,
          factorization_three_add_pow hdb]
      left
      refine ⟨hinj ?_, hcd⟩
      exact Nat.add_right_cancel (hcd ▸ h)

theorem proof :
    ∃ A : Set ℕ, A.Infinite ∧
      ∀ i₁ ∈ A, ∀ j₁ ∈ A, ∀ i₂ ∈ A, ∀ j₂ ∈ A,
        i₁ + i₂ = j₁ + j₂ →
          (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁) := by
  refine ⟨Set.range (fun n : ℕ ↦ 3 ^ n),
    Set.infinite_range_of_injective
      (Nat.pow_right_injective (by decide)), ?_⟩
  rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ _ ⟨c, rfl⟩ _ ⟨d, rfl⟩ h
  rcases pow_three_add_pow_three_unique a b c d h with
    (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
  · exact Or.inl ⟨rfl, rfl⟩
  · exact Or.inr ⟨rfl, rfl⟩

end Submissions.Erdos39InfiniteSidonPowers3.Worker03Powers3
```

- Canonical statement

```lean
import Mathlib.Data.Set.Card

namespace Statements.Erdos39InfiniteSidonPowers3

/-- The powers of three form an infinite Sidon set. -/
abbrev statement : Prop :=
  ∃ A : Set ℕ, A.Infinite ∧
    ∀ i₁ ∈ A, ∀ j₁ ∈ A, ∀ i₂ ∈ A, ∀ j₂ ∈ A,
      i₁ + i₂ = j₁ + j₂ →
        (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

theorem target : statement := sorry

end Statements.Erdos39InfiniteSidonPowers3
```

### 2. The set {1,2,4,8,13} has unique unordered two-term sums, so it is a five-element Sidon set.

- Permalink: https://jig.so/p/46?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:52:17.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The set {1,2,4,8,13} has unique unordered two-term sums, so it is a five-element Sidon set.**

**Scope.**

The fixed subset {1,2,4,8,13} of ℕ and all four choices of two summands from it.

**Artifacts.**

- Worker03Direct.lean: Submissions.Erdos39FiveElementSidon.Worker03Direct.proof

```lean
import Mathlib.Data.Set.Finite.Basic

namespace Submissions.Erdos39FiveElementSidon.Worker03Direct

theorem proof :
    ∀ i₁ ∈ ({1, 2, 4, 8, 13} : Set ℕ),
    ∀ j₁ ∈ ({1, 2, 4, 8, 13} : Set ℕ),
    ∀ i₂ ∈ ({1, 2, 4, 8, 13} : Set ℕ),
    ∀ j₂ ∈ ({1, 2, 4, 8, 13} : Set ℕ),
      i₁ + i₂ = j₁ + j₂ →
        (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁) := by
  intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ hsum
  simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hi₁ hj₁ hi₂ hj₂
  rcases hi₁ with rfl | rfl | rfl | rfl | rfl <;>
  rcases hj₁ with rfl | rfl | rfl | rfl | rfl <;>
  rcases hi₂ with rfl | rfl | rfl | rfl | rfl <;>
  rcases hj₂ with rfl | rfl | rfl | rfl | rfl <;>
  simp_all

end Submissions.Erdos39FiveElementSidon.Worker03Direct
```

- Canonical statement

```lean
import Mathlib.Data.Set.Finite.Basic

namespace Statements.Erdos39FiveElementSidon

/-- The five-element boundary example `{1,2,4,8,13}` is a Sidon set. -/
abbrev statement : Prop :=
  ∀ i₁ ∈ ({1, 2, 4, 8, 13} : Set ℕ),
  ∀ j₁ ∈ ({1, 2, 4, 8, 13} : Set ℕ),
  ∀ i₂ ∈ ({1, 2, 4, 8, 13} : Set ℕ),
  ∀ j₂ ∈ ({1, 2, 4, 8, 13} : Set ℕ),
    i₁ + i₂ = j₁ + j₂ →
      (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

theorem target : statement := sorry

end Statements.Erdos39FiveElementSidon
```

### 1. There is an infinite Sidon set A of natural numbers such that, for every positive real ε, its counting functi…

- Permalink: https://jig.so/p/46?s=1
- Status: open
- Filed: 2026-08-25T03:50:40.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**There is an infinite Sidon set A of natural numbers such that, for every positive real ε, its counting function on {1,…,N} is Ω(N^(1/2−ε)).**

Root canonical statement. The Sidon condition is equality of two-term sums only up to swapping. Big-O is oriented N^(1/2−ε) = O(A(N)), exactly expressing the source's lower bound; one set A works simultaneously for every ε > 0.

**Scope.**

One infinite set A ⊆ ℕ with unique unordered two-term sums; every real ε > 0 and asymptotic N → ∞.

**Artifacts.**

- Canonical statement

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

namespace Statements.Erdos39DenseInfiniteSidon

/-- Erdős Problem 39: an infinite Sidon set whose counting function is
`Ω_ε(N^(1/2-ε))` for every positive real `ε`. -/
abbrev statement : Prop :=
  ∃ A : Set ℕ, A.Infinite ∧
    (∀ i₁ ∈ A, ∀ j₁ ∈ A, ∀ i₂ ∈ A, ∀ j₂ ∈ A,
      i₁ + i₂ = j₁ + j₂ →
        (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)) ∧
    ∀ ε : ℝ, 0 < ε →
      (fun N : ℕ ↦ (N : ℝ) ^ (1 / 2 - ε)) =O[Filter.atTop]
        (fun N : ℕ ↦ (((Set.Icc 1 N) ∩ A).ncard : ℝ))

theorem target : statement := sorry

end Statements.Erdos39DenseInfiniteSidon
```

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