828.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, 3 contain sorry, 0 declared as axioms

Erdős Problem 828

Reference: erdosproblems.com/828

Imports (1)

FormalConjecturesUtil

theorem Erdos828.erdos_828 Is it true that, for any $a \in \mathbb{Z}$, there are infinitely many $n$ such that $$\phi(n) | n + a$$? sorry line 37 source
Source documentation
Is it true that, for any $a \in \mathbb{Z}$, there are infinitely many $n$ such that $$\phi(n) | n + a$$?

depends on sorry what this means
Statement uses: Iff, True, Int, Set.Infinite, Nat, Set.ofPred, Dvd.dvd, Nat.totient, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_828 : answer(sorry) ↔ ∀ a : ℤ, Set.Infinite {n : ℕ | ↑(φ n) ∣ n + a} := by
…
```
theorem Erdos828.erdos_828.variants.lehmer_conjecture When $n > 1$, Lehmer conjectured that $\phi(n) | n - 1$ if and only if $n$ is prime sorry line 44 source
Source documentation
When $n > 1$, Lehmer conjectured that $\phi(n) | n - 1$ if and only if $n$ is prime.

depends on sorry what this means
Statement uses: Iff, True, Nat, GT.gt, OfNat.ofNat, Dvd.dvd, Nat.totient, HSub.hSub, Prime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_828.variants.lehmer_conjecture : answer(sorry) ↔ ∀ n > 1, φ n ∣ n - 1 ↔ Prime n := by
…
```
theorem Erdos828.erdos_828.variants.phi_dvd_self_iff_pow2_pow3 sorry line 53 source
depends on sorry what this means
Statement uses: Nat, Iff, Dvd.dvd, Nat.totient, Or, LE.le, OfNat.ofNat, Exists, And, GT.gt, Eq, HMul.hMul, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.