1134.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1134

References:

erdosproblems.com/1134
[La16] Lagarias, Jeffrey C., Erdős, Klarner, and the $3x+1$ problem. Amer. Math. Monthly (2016), 753--776.
[KlRa74] Klarner, D. A. and Rado, R., Arithmetic properties of certain recursively defined sets. Pacific J. Math. (1974), 445--463.
[Kl82] Klarner, David A., A sufficient condition for certain semigroups to be free. J. Algebra (1982), 140--148.
[Gu83b] Guy, Richard K., Unsolved Problems: Don't Try to Solve These Problems. Amer. Math. Monthly (1983), 35--38+39--41.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

Imports (1)

FormalConjecturesUtil

def Erdos1134.A The smallest set of natural numbers which contains 1 and is closed under the operations x ↦ 2x + 1, x ↦ 3x + 1 and x ↦ 6x + 1 line 42 source
Source documentation
The smallest set of natural numbers which contains 1 and is closed under the operations x ↦ 2x + 1, x ↦ 3x + 1 and x ↦ 6x + 1.

no axioms what this means
Statement uses: Set, Nat
Proof uses: Set.sInter, Set.ofPred, And, Membership.mem, OfNat.ofNat, HAdd.hAdd, HMul.hMul
6 instances, coercions and auxiliary constants left out.
```
def A : Set ℕ :=
…
```
theorem Erdos1134.erdos_1134 sorry line 70 source
depends on sorry what this means
Statement uses: Iff, False, LT.lt, Real, OfNat.ofNat, Set.lowerDensity, Nat, Erdos1134.A, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1134.erdos_1134.variants.sublinear sorry line 77 source
depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Set.ncard, Inter.inter, Set, Erdos1134.A, Set.Iic, HMul.hMul, HPow.hPow, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1134.erdos_1134.variants.klarner sorry line 92 source
depends on sorry what this means
Statement uses: Iff, False, LT.lt, Real, OfNat.ofNat, Set.lowerDensity, Nat, Set.sInter, Set.ofPred, Set, And, Membership.mem, HMul.hMul, HAdd.hAdd, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.