419.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

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 419

References:

erdosproblems.com/419
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[EGIP96] Erdős, Paul and Graham, S. W. and Ivić, Aleksandar and Pomerance, Carl, On the number of divisors of $n!$. (1996), 337--355.

Imports (1)

FormalConjecturesUtil

def Erdos419.factorialDivisorRatio The ratio $\sigma_0((n+1)!)/\sigma_0(n!)$, where $\sigma_0$ is the divisor-counting function line 39 source
Source documentation
The ratio $\sigma_0((n+1)!)/\sigma_0(n!)$, where $\sigma_0$ is the divisor-counting function.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HDiv.hDiv, ArithmeticFunction, ArithmeticFunction.sigma, OfNat.ofNat, Nat.factorial, HAdd.hAdd
12 instances, coercions and auxiliary constants left out.
```
noncomputable def factorialDivisorRatio (n : ℕ) : ℝ :=
…
```
def Erdos419.limitPointSet The set $\{1\} \cup \{1+1/k : k \geq 1\}$ line 43 source
Source documentation
The set $\{1\} \cup \{1+1/k : k \geq 1\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real
Proof uses: Union.union, Singleton.singleton, OfNat.ofNat, Set.ofPred, Exists, Nat, And, LE.le, Eq, HAdd.hAdd, HDiv.hDiv
13 instances, coercions and auxiliary constants left out.
```
def limitPointSet : Set ℝ :=
…
```
theorem Erdos419.erdos_419 Documented declaration: erdos_419 sorry line 55 source
Source documentation
If $\tau(n)$ counts the number of divisors of $n$, then what is the set of limit points of $$ \frac{\tau((n+1)!)}{\tau(n!)}? $$
The limit points are exactly $\{1\} \cup \{1+1/k : k \geq 1\}$.

depends on sorry what this means
Statement uses: Eq, Set, Real, Set.ofPred, MapClusterPt, Nat, Filter.atTop, Erdos419.factorialDivisorRatio, Erdos419.limitPointSet
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 erdos_419 :
    {x : ℝ | MapClusterPt x atTop factorialDivisorRatio} = limitPointSet := by
…
```
Mentions: Erdos419.factorialDivisorRatio, Erdos419.limitPointSet