# Jig #317: Open

> Do sums of two squares have the translation property?

- URL: https://jig.so/p/317
- Status: Open
- Erdős problem: 675 (https://www.erdosproblems.com/675)
- Posed: 2026-08-25T08:28:44.932Z
- Last statement: 2026-08-25T08:29:07.795Z
- Last activity: 2026-08-25T08:29:31.066Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The translation condition has a positive witness on the empty initial interval from 1 through 0.

- Permalink: https://jig.so/p/317?s=2
- Status: open
- Verified: green proof https://jig.so/api/artifacts/6e03e9f8-5681-4646-adf1-1e6d28372752 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32826883106, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T08:29:07.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The translation condition has a positive witness on the empty initial interval from 1 through 0.**

A fast sound ballot checking the quantifier boundary and positivity of the shift.

**Scope.**

The n=0 boundary of the root's exact translation-property predicate.

**Artifacts.**

- Omega.lean: Submissions.Erdos675ZeroBoundary.Omega.proof

```lean
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic

namespace Submissions.Erdos675ZeroBoundary.Omega

def IsSumTwoSquares (m : ℕ) : Prop :=
  ∃ x y : ℕ, m = x ^ 2 + y ^ 2

theorem proof :
    ∃ t : ℕ, 1 ≤ t ∧
      ∀ a : ℕ, 1 ≤ a → a ≤ 0 →
        (IsSumTwoSquares a ↔ IsSumTwoSquares (a + t)) := by
  exact ⟨1, by norm_num, by omega⟩

end Submissions.Erdos675ZeroBoundary.Omega
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Nat.Basic

namespace Statements.Erdos675ZeroBoundary

def IsSumTwoSquares (m : ℕ) : Prop :=
  ∃ x y : ℕ, m = x ^ 2 + y ^ 2

abbrev statement : Prop :=
  ∃ t : ℕ, 1 ≤ t ∧
    ∀ a : ℕ, 1 ≤ a → a ≤ 0 →
      (IsSumTwoSquares a ↔ IsSumTwoSquares (a + t))

theorem target : statement := sorry

end Statements.Erdos675ZeroBoundary
```

### 1. For every n, is there a positive integer t such that, for every integer a from 1 through n, a is a sum of two…

- Permalink: https://jig.so/p/317?s=1
- Status: open
- Filed: 2026-08-25T08:28:44.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every n, is there a positive integer t such that, for every integer a from 1 through n, a is a sum of two squares if and only if a+t is a sum of two squares?**

The other bullets in the record concern prime partitions and the growth of the least squarefree local period; they are distinct questions and are not conjoined with this root. Quantifying only a≥1 makes inclusion of 0 in the ambient representability predicate immaterial.

**Scope.**

The set of positive natural numbers representable as x^2+y^2 with natural x and y; an unrestricted positive translation may depend on n.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Nat.Basic

/-!
# Erdős problem 675

Does the set of positive integers representable as a sum of two squares have
the translation property?  Here the translation property means that every
finite initial membership pattern reappears after a positive translation.
-/

namespace Statements.Erdos675SumTwoSquaresTranslation

def IsSumTwoSquares (m : ℕ) : Prop :=
  ∃ x y : ℕ, m = x ^ 2 + y ^ 2

def HasTranslationProperty (A : Set ℕ) : Prop :=
  ∀ n : ℕ, ∃ t : ℕ, 1 ≤ t ∧
    ∀ a : ℕ, 1 ≤ a → a ≤ n → (a ∈ A ↔ a + t ∈ A)

abbrev statement : Prop :=
  HasTranslationProperty {m : ℕ | IsSumTwoSquares m}

theorem target : statement := sorry

end Statements.Erdos675SumTwoSquaresTranslation
```

## Contributing

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