478.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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 478

References:

erdosproblems.com/478
[AnTa16] V. Andrejić and M. Tatarevic, On distinct residues of factorials. arXiv:1603.04086 (2016).
[GSSV24] Grebennikov, Alexandr and Sagdeev, Arsenii and Semchankau, Aliaksei and Vasilevskii, Aliaksei, On the sequence {$n! \bmod p$}. Rev. Mat. Iberoam. (2024), 637--648.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[KlMu17] Klurman, Oleksiy and Munsch, Marc, Distribution of factorials modulo {$p$}. J. Théor. Nombres Bordeaux (2017), 169--177.
[RoSc60] Rokowska, B. and Schinzel, A., Sur un problème de {M}. {E}rdős. Elem. Math. (1960), 84--85.
[Tr13] T. Trudgian, There are no socialist primes less than $10^9$. arXiv:1310.6403 (2013).

Imports (1)

FormalConjecturesUtil

theorem Erdos478.erdos_478 Let $p$ be a prime and $$A_p = \{ k! \pmod{p} : 1\leq k<p\}.$$ Is it true that $$\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?$$ sorry line 45 source
Source documentation
Let $p$ be a prime and $$A_p = \{ k! \pmod{p} : 1\leq k<p\}.$$ Is it true that $$\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?$$

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, Real, HDiv.hDiv, Finset.card, Finset.image, HMod.hMod, Nat.factorial, Finset.Ico, OfNat.ofNat, Min.min, Filter, Filter.atTop, Filter.principal, Set.ofPred, Nat.Prime, nhds, HSub.hSub and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_478 : answer(sorry) ↔
    Filter.Tendsto
      (fun p : ℕ =>
        (((Finset.Ico 1 p).image (fun k => Nat.factorial k % p)).card : ℝ) / p)
      (Filter.atTop ⊓ Filter.principal {p : ℕ | p.Prime})
      (nhds (1 - 1 / Real.exp 1)) := by
…
```