822.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 822

References:

erdosproblems.com/822
[GIL24] Gabdullin, Mikhail R. and Iudelevich, Vitalii V. and Luca, Florian, Numbers of the form {$k+f(k)$}. J. Number Theory (2024), 58--85.

Imports (1)

FormalConjecturesUtil

theorem Erdos822.erdos_822 Does the set of integers of the form $n + \varphi(n)$ have positive (lower) density? sorry line 39 source
Source documentation
Does the set of integers of the form $n + \varphi(n)$ have positive (lower) density?
[GIL24] proved this was true.

depends on sorry what this means
Statement uses: Iff, True, LT.lt, Real, OfNat.ofNat, Set.lowerDensity, Nat, Set.range, HAdd.hAdd, Nat.totient, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_822 :
    answer(True) ↔ 0 < (Set.range fun n => n + Nat.totient n).lowerDensity := by
…
```