433.lean`

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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

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 433

References:

Erdős Problem 433
P. Erdős and R. L. Graham, On a linear Diophantine problem of Frobenius, Acta Arithmetica 21 (1972), 399–408.
J. Dixmier, Proof of a conjecture by Erdős and Graham on Frobenius' coin problem, Proceedings of the AMS 109 (1990), 567–577.

Imports (1)

FormalConjecturesUtil

def Erdos433.frobeniusNumber The Frobenius number of a finite set $A$: the greatest natural number outside the additive submonoid generated by $A$ line 40 source
Source documentation
The Frobenius number of a finite set $A$: the greatest natural number outside the additive submonoid generated by $A$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: SupSet.sSup, Set.ofPred, Not, Membership.mem, AddSubmonoid, AddSubmonoid.closure
7 instances, coercions and auxiliary constants left out.
```
noncomputable def frobeniusNumber (A : Finset ℕ) : ℕ :=
…
```
def Erdos433.g The largest Frobenius number of a $k$-element subset of $\{1,\ldots,n\}$ whose elements have greatest common divisor one line 45 source
Source documentation
The largest Frobenius number of a $k$-element subset of $\{1,\ldots,n\}$ whose elements have greatest common divisor one.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Eq, Finset.card, Finset.gcd, id, Erdos433.frobeniusNumber
10 instances, coercions and auxiliary constants left out.
```
noncomputable def g (k n : ℕ) : ℕ :=
…
```
theorem Erdos433.erdos_433 For every fixed $k \geq 2$, Erdős and Graham conjectured that $$g(k,n) \sim \frac{n^2}{k-1}.$$ The conjecture was proved by Dixmier [Di90] sorry line 53 source
Source documentation
For every fixed $k \geq 2$, Erdős and Graham conjectured that $$g(k,n) \sim \frac{n^2}{k-1}.$$ The conjecture was proved by Dixmier [Di90].

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Filter.Tendsto, Real, HDiv.hDiv, Erdos433.g, HPow.hPow, HSub.hSub, Filter.atTop, nhds
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 erdos_433 : answer(True) ↔ ∀ k, 2 ≤ k →
    Tendsto (fun n : ℕ => (g k n : ℝ) / ((n : ℝ) ^ 2 / ((k : ℝ) - 1))) atTop (𝓝 1) := by
…
```
Mentions: Erdos433.g