429.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 429

References:

erdosproblems.com/429
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[We24] D. Weisenberg, Sparse Admissible Sets and a Problem of Erdős and Graham. Integers (2024).

Imports (1)

FormalConjecturesUtil

theorem Erdos429.erdos_429 Documented declaration: erdos_429 sorry line 49 source
Source documentation
Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n\in \mathbb{Z}$ such that $n+a$ is prime for all $a\in A$?
Weisenberg [We24] has shown the answer is no: $A$ can be arbitrarily sparse and missing at least one residue class modulo every prime $p$, and yet $A+n$ is not contained in the primes for any $n\in \mathbb{Z}$. (Weisenberg gives several constructions of such an $A$.)

depends on sorry what this means
Statement uses: Iff, False, Exists, Nat, And, Filter.Tendsto, Filter.atTop, Set, Set.Infinite, LE.le, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Nat.Prime, ZMod, Membership.mem, Ne, Int, Int.toNat and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_429 : answer(False) ↔
    ∃ f : ℕ → ℕ, Tendsto f atTop atTop ∧
      ∀ A : Set ℕ, A.Infinite → (∀ N, (A ∩ Set.Icc 1 N).ncard ≤ f N) →
        (∀ p : ℕ, p.Prime → ∃ b : ZMod p, ∀ a ∈ A, (a : ZMod p) ≠ b) →
        ∃ n : ℤ, ∀ a ∈ A, (n + a).toNat.Prime := by
…
```