SolitaryNumber.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

4 declarations, 2 contain sorry, 0 declared as axioms

Solitary Numbers

References:

Imports (1)

FormalConjecturesUtil

def SolitaryNumber.Friendly Two positive integers line 39 source
Source documentation
Two positive integers $m$ and $n$ are friendly if they have the same abundancy index, that is $\sigma(m) / m = \sigma(n) / n$, expressed via cross-multiplication to avoid rationals.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, LT.lt, OfNat.ofNat, Eq, HMul.hMul, ArithmeticFunction, ArithmeticFunction.sigma
8 instances, coercions and auxiliary constants left out.
```
def Friendly (m n : ℕ) : Prop := 0 < m ∧ 0 < n ∧ σ 1 m * n = σ 1 n * m
```
def SolitaryNumber.IsSolitary A positive integer $n$ is solitary if every friend of $n$ is equal to $n$, i.e. its abundancy class is the singleton $\{n\}$ line 45 source
Source documentation
A positive integer $n$ is solitary if every friend of $n$ is equal to $n$, i.e. its abundancy class is the singleton $\{n\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, LT.lt, OfNat.ofNat, SolitaryNumber.Friendly, Eq
2 instances, coercions and auxiliary constants left out.
```
def IsSolitary (n : ℕ) : Prop := 0 < n ∧ ∀ m, Friendly m n → m = n
…
```
Mentions: SolitaryNumber.Friendly
theorem SolitaryNumber.is_ten_solitary Is 10 a solitary number? sorry line 52 source
Source documentation
Is 10 a solitary number? The smallest positive integer whose solitary status is currently unresolved is $10$, with abundancy index $\sigma(10) / 10 = 9/5$.

depends on sorry what this means
Statement uses: Iff, True, SolitaryNumber.IsSolitary, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem is_ten_solitary : answer(sorry) ↔ IsSolitary 10 := by
…
```
Mentions: SolitaryNumber.IsSolitary
theorem SolitaryNumber.infinite_club_exists Existence of an infinite club sorry line 61 source
Source documentation
Existence of an infinite club. A club is an abundancy equivalence class, i.e. the set of all positive integers friendly with a given $n$. It is unknown whether any club is infinite.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, LT.lt, OfNat.ofNat, Set.Infinite, Set.ofPred, SolitaryNumber.Friendly
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 infinite_club_exists :
    answer(sorry) ↔ ∃ n, 0 < n ∧ {m : ℕ | Friendly m n}.Infinite := by
…
```
Mentions: SolitaryNumber.Friendly