793.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.

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

4 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 793

References:

erdosproblems.com/793
[Er38] Erdős, P., On sequences of integers no one of which divides the product of two others and on related problems. Tomsk. Gos. Univ. Ucen Zap. (1938), 74-82.
[Er69] Erdős, Paul, Some applications of graph theory to number theory. The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968) (1969), 77-82.
[Er70b] Erdős, P., Some applications of graph theory to number theory. Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970) (1970), 136-145.

Imports (1)

FormalConjecturesUtil

def Erdos793.Strongly2Primitive A finite set A ⊆ ℕ is strongly 2-primitive if a ∤ b * c whenever a, b, c ∈ A with a ≠ b and a ≠ c line 44 source
Source documentation
A finite set A ⊆ ℕ is strongly 2-primitive if a ∤ b * c whenever a, b, c ∈ A with a ≠ b and a ≠ c.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Membership.mem, Ne, Not, Dvd.dvd, HMul.hMul
5 instances, coercions and auxiliary constants left out.
```
def Strongly2Primitive (A : Finset ℕ) : Prop :=
…
```
def Erdos793.F F n is the maximal size of a strongly 2-primitive subset of {1, …, n} line 48 source
Source documentation
F n is the maximal size of a strongly 2-primitive subset of {1, …, n}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sup, Finset, Finset.filter, Erdos793.Strongly2Primitive, Classical.propDecidable, Finset.powerset, Finset.Icc, OfNat.ofNat, Finset.card
6 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
noncomputable def F (n : ℕ) : ℕ := by
…
```
theorem Erdos793.erdos_793 Let $F(n)$ be the maximum possible size of a subset $A \subseteq \{1, \ldots, n\}$ such that $a \nmid bc$ whenever $a, b, c \in A$ with $a \neq b$ and $a \neq c$ sorry line 64 source
Source documentation
Let $F(n)$ be the maximum possible size of a subset $A \subseteq \{1, \ldots, n\}$ such that $a \nmid bc$ whenever $a, b, c \in A$ with $a \neq b$ and $a \neq c$. Is there a constant $c$ such that $$F(n) = \pi(n) + (c + o(1)) n^{2/3} (\log n)^{-2}?$$
A problem of Erdős [Er69, Er70b], who proved in [Er38] that $F(n) = \pi(n) + O(n^{2/3} (\log n)^{-2})$. The answer is yes, with $c = 27/2$: this was proved by GPT-5.6 Sol (prompted by Chojecki), refining the argument of [Er38]; see erdos_793.variants.constant.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, Filter.Tendsto, Nat, HDiv.hDiv, HSub.hSub, Erdos793.F, Nat.primeCounting, HPow.hPow, OfNat.ofNat, Real.log, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_793 : answer(True) ↔
    ∃ c : ℝ, Tendsto (fun n : ℕ ↦ ((F n : ℝ) - Nat.primeCounting n) /
      ((n : ℝ) ^ ((2 : ℝ) / 3) / (log n) ^ 2)) atTop (𝓝 c) := by
…
```
Mentions: c, Erdos793.F
theorem Erdos793.erdos_793.variants.constant sorry line 76 source
depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, HSub.hSub, Erdos793.F, Nat.primeCounting, HPow.hPow, OfNat.ofNat, Real.log, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

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