Erdős (Erdos) Problem #675: status in 2026

Erdős Problem #675

A set A ⊆ ℕ has the translation property if, for every n, there is an integer t_n ≥ 1 such that, for all 1 ≤ a ≤ n, a ∈ A if and only if a + t_n ∈ A. Does the set of sums of two squares x^2 + y^2 : x, y ∈ ℕ have the translation property?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #675 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #675 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 3 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-erdos-675,
  title        = {Erdős Problem #675},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-675}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

erdos_675.parts.i. A set $A \subseteq N$ has the translation property if, for every $n$ , there is an integer $t_{n} \geq 1$ such that, for all $1 \leq a \leq n$ , $a \in A$ if and only if $a + t_{n} \in A$ . Does the set of sums of two squares ${x^{2} + y^{2} : x, y \in N}$ have the translation property?

erdos_675.parts.ii. If we partition all primes into $P ⊔ Q$ , such that each set contains $≫ x / lo g x$ many primes $\leq x$ for all large $x$ , then can the set of integers only divisible by primes from $P$ have the translation property?

We read "can" as: is there such a partition for which this set (which contains $1$ ) has the translation property?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«675» (2 statements).

theorem erdos_675.parts.i :
    HasTranslationProperty {m : ℕ | ∃ x y : ℕ, m = x ^ 2 + y ^ 2}

theorem erdos_675.parts.ii :
    ∃ P : Set ℕ, P ⊆ {p : ℕ | p.Prime} ∧
      (fun x : ℕ ↦ (x : ℝ) / Real.log x) =O[atTop] (fun x ↦ ((P ∩ Set.Iic x).ncard : ℝ)) ∧
      (fun x : ℕ ↦ (x : ℝ) / Real.log x) =O[atTop]
        (fun x ↦ ((({p : ℕ | p.Prime} \ P) ∩ Set.Iic x).ncard : ℝ)) ∧
      HasTranslationProperty (composedOf P)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/675. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/675
[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
[Ho26] Ho, B. S., A squarefree lower bound for Erdős Problem 675. PDF (2026).

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.