858.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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 858

References:

erdosproblems.com/858
[Er70] Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.
[Al66] Alexander, Ralph, Density and multiplicative structure of sets of integers. Acta Arith. (1966/67), 321--332.
[ESS68] Erdős, P. and Sárközi, A. and Szemerédi, E., On the solvability of certain equations in sequences of positive upper logarithmic density. J. London Math. Soc. (1968), 71--78.
[Be35] Behrend, F., On sequences of numbers not divisible by another. London Math. Soc. Journal (1935), 42-45.

Imports (1)

FormalConjecturesUtil

def Erdos858.IsAdmissible A has no solution to $at=b$ with $a,b\in A$ and the smallest prime factor of $t$ greater than $a$ line 43 source
Source documentation
A has no solution to $at=b$ with $a,b\in A$ and the smallest prime factor of $t$ greater than $a$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Membership.mem, Eq, HMul.hMul, LE.le, Nat.minFac
5 instances, coercions and auxiliary constants left out.
```
def IsAdmissible (A : Finset ℕ) : Prop :=
…
```
def Erdos858.M The maximum of $\sum_{n\in A}\frac{1}{n}$ over all admissible $A\subseteq \{1,\ldots,N\}$ line 47 source
Source documentation
The maximum of $\sum_{n\in A}\frac{1}{n}$ over all admissible $A\subseteq \{1,\ldots,N\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Erdos858.IsAdmissible, Eq, Finset.sum, HDiv.hDiv
15 instances, coercions and auxiliary constants left out.
```
noncomputable def M (N : ℕ) : ℝ :=
…
```
theorem Erdos858.erdos_858 sorry line 62 source
depends on sorry what this means
Statement uses: Exists, Real, Filter.Tendsto, Nat, HDiv.hDiv, Erdos858.M, Real.log, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos858.erdos_858.variants.positive The limit $c\approx 0.618\cdots$ is positive: for any fixed large $N$ the maximum is bounded away from $0$ (Alexander [Al66], Erdős–Sárközi–Szemerédi [ESS68]) sorry line 68 source
Source documentation
The limit $c\approx 0.618\cdots$ is positive: for any fixed large $N$ the maximum is bounded away from $0$ (Alexander [Al66], Erdős–Sárközi–Szemerédi [ESS68]).

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Tendsto, Nat, HDiv.hDiv, Erdos858.M, Real.log, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_858.variants.positive :
    ∃ c : ℝ, 0 < c ∧ Tendsto (fun N : ℕ ↦ M N / log N) atTop (nhds c) := by
…
```
Mentions: c, Erdos858.M
theorem Erdos858.erdos_858.variants.infinite Alexander [Al66] and Erdős, Sárközi, and Szemerédi [ESS68] proved that if $A$ is an infinite set with this property then $\sum_{n\in A\cap [1,N]}\frac{1}{n}=o(\log N)$ sorry line 78 source
Source documentation
Alexander [Al66] and Erdős, Sárközi, and Szemerédi [ESS68] proved that if $A$ is an infinite set with this property then $\sum_{n\in A\cap [1,N]}\frac{1}{n}=o(\log N)$.

depends on sorry what this means
Statement uses: Set, Nat, Set.Infinite, Not, Membership.mem, OfNat.ofNat, Eq, HMul.hMul, LE.le, Nat.minFac, Asymptotics.IsLittleO, Real, Filter.atTop, Finset.sum, Finset.filter, Classical.propDecidable, Finset.Icc, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_858.variants.infinite (A : Set ℕ) (hA : A.Infinite) (h0 : 0 ∉ A)
    (hadm : ∀ a ∈ A, ∀ b ∈ A, ∀ t : ℕ, a * t = b → t.minFac ≤ a) :
    (fun N : ℕ ↦ ∑ n ∈ (Finset.Icc 1 N).filter (· ∈ A), (1 : ℝ) / n) =o[atTop]
      fun N : ℕ ↦ log N := by
…
```