692.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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

8 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 692

References:

erdosproblems.com/692
[Er79e] Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82.
[Ob1] Erdős, P., Oberwolfach Mathematical Problems, Volume 1. Mathematisches Forschungsinstitut Oberwolfach (Various).
[Fo08] Ford, Kevin, The distribution of integers with a divisor in a given interval. Ann. of Math. (2) (2008), 367-433.
[Ca25] Cambie, S., Resolution of Erdős' problems about unimodularity. arXiv:2501.10333 (2025).

Imports (1)

FormalConjecturesUtil

def Erdos692.exactlyOneDivisorIn The set of integers with exactly one divisor in the open interval $(n, m)$ line 43 source
Source documentation
The set of integers with exactly one divisor in the open interval $(n, m)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Eq, Set.ncard, And, Membership.mem, Set.Ioo, Dvd.dvd, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
```
def exactlyOneDivisorIn (n m : ℕ) : Set ℕ :=
…
```
def Erdos692.IsDelta₁ $\delta_1(n,m)$ is the density of the set of integers with exactly one divisor in $(n,m)$ line 49 source
Source documentation
$\delta_1(n,m)$ is the density of the set of integers with exactly one divisor in $(n,m)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Set.HasDensity, Erdos692.exactlyOneDivisorIn, Set.univ
4 instances, coercions and auxiliary constants left out.
```
def IsDelta₁ (n m : ℕ) (δ : ℝ) : Prop :=
…
```
def Erdos692.localMaxima The set of local maxima of the sequence f on (a, ∞) line 57 source
Source documentation
The set of local maxima of the sequence f on (a, ∞). Each local maximum is recorded by the first index m of a maximal interval [m, b] on which f is constant, with f strictly smaller at m - 1 and at b + 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Set
Proof uses: Set.ofPred, And, LT.lt, HSub.hSub, OfNat.ofNat, Exists, LE.le, Membership.mem, Set.Icc, Eq, HAdd.hAdd
10 instances, coercions and auxiliary constants left out.
```
def localMaxima (f : ℕ → ℝ) (a : ℕ) : Set ℕ :=
…
```
theorem Erdos692.erdos_692.parts.i sorry line 71 source
depends on sorry what this means
Statement uses: Iff, False, Nat, Real, Erdos692.IsDelta₁, UnimodularOn, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos692.erdos_692.parts.ii Let $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$ sorry line 80 source
Source documentation
Let $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. For fixed $n$, where does $\delta_1(n,m)$ achieve its maximum?

depends on sorry what this means
Statement uses: Nat, Real, Erdos692.IsDelta₁, IsGreatest, Set.image, Set.Ioi, HAdd.hAdd, OfNat.ofNat, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_692.parts.ii (n : ℕ) :
    ∀ δ : ℕ → ℕ → ℝ, (∀ a b, IsDelta₁ a b (δ a b)) →
      IsGreatest (δ n '' Set.Ioi (n + 1)) (δ n answer(sorry)) := by
…
```
Mentions: Erdos692.IsDelta₁
theorem Erdos692.erdos_692.variants.erdos_upper_bound Erdős proved that $$\delta_1(n,m) \ll \frac{1}{(\log n)^c}$$ for all $m$, for some constant $c>0$ sorry line 91 source
Source documentation
Erdős proved that $$\delta_1(n,m) \ll \frac{1}{(\log n)^c}$$ for all $m$, for some constant $c>0$.

depends on sorry what this means
Statement uses: Nat, Real, Erdos692.IsDelta₁, Exists, And, GT.gt, OfNat.ofNat, Filter.Eventually, LE.le, HDiv.hDiv, HPow.hPow, Real.log, Filter.atTop
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_692.variants.erdos_upper_bound :
    ∀ δ : ℕ → ℕ → ℝ, (∀ a b, IsDelta₁ a b (δ a b)) →
      ∃ c > (0 : ℝ), ∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∀ m, δ n m ≤ C / Real.log n ^ c := by
…
```
Mentions: Erdos692.IsDelta₁, c
theorem Erdos692.erdos_692.variants.cambie_three Cambie has calculated that unimodularity fails even for $n=2$ and $n=3$ sorry line 101 source
Source documentation
Cambie has calculated that unimodularity fails even for $n=2$ and $n=3$. For example, $$\delta_1(3,6)= 0.35\quad \delta_1(3,7)\approx 0.33\quad \delta_1(3,8)\approx 0.3619.$$

depends on sorry what this means
Statement uses: Nat, Real, Erdos692.IsDelta₁, And, LT.lt, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_692.variants.cambie_three :
    ∀ δ : ℕ → ℕ → ℝ, (∀ a b, IsDelta₁ a b (δ a b)) →
      δ 3 7 < δ 3 6 ∧ δ 3 7 < δ 3 8 := by
…
```
Mentions: Erdos692.IsDelta₁
theorem Erdos692.erdos_692.variants.cambie_local_maxima Cambie [Ca25] has shown that, for fixed $n$, the sequence $\delta_1(n,m)$ has superpolynomially many local maxima $m$ sorry line 112 source
Source documentation
Cambie [Ca25] has shown that, for fixed $n$, the sequence $\delta_1(n,m)$ has superpolynomially many local maxima $m$. A local maximum is a maximal run of equal values of the sequence that is strictly larger than the values just before and just after the run.

depends on sorry what this means
Statement uses: Nat, Real, Erdos692.IsDelta₁, Filter.Eventually, LE.le, ENat, HPow.hPow, Set.encard, Erdos692.localMaxima, HAdd.hAdd, OfNat.ofNat, Filter.atTop
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.
```
theorem erdos_692.variants.cambie_local_maxima (k : ℕ) :
    ∀ δ : ℕ → ℕ → ℝ, (∀ a b, IsDelta₁ a b (δ a b)) →
      ∀ᶠ n : ℕ in atTop, (n : ℕ∞) ^ k ≤ (localMaxima (δ n) (n + 1)).encard := by
…
```
Mentions: Erdos692.IsDelta₁, Erdos692.localMaxima