309.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 309

References:

erdosproblems.com/309
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Yo97] Yokota, Hisashi, On number of integers representable as a sum of unit fractions. II. J. Number Theory (1997), 162--169.
[Cr99] Croot, III, Ernest S., On some questions of Erdős and Graham about Egyptian fractions. Mathematika (1999), 359-372.
[Yo02] Yokota, Hisashi, On the number of integers representable as sums of unit fractions. III. J. Number Theory (2002), 351--372.

Imports (1)

FormalConjecturesUtil

def Erdos309.F The number of integers which can be written as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$ line 43 source
Source documentation
The number of integers which can be written as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set.ncard, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Eq, Rat, Finset.sum, HDiv.hDiv
12 instances, coercions and auxiliary constants left out.
```
noncomputable def F (N : ℕ) : ℕ :=
…
```
theorem Erdos309.erdos_309 sorry line 66 source
depends on sorry what this means
Statement uses: Iff, False, Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos309.F, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos309.erdos_309.variants.asymptotic sorry line 73 source
depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Erdos309.F, Real.log, Filter.atTop, nhds, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
23 instances, coercions and auxiliary constants left out.

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