830.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

6 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 830

References:

erdosproblems.com/830
[Er55b] Erdős, P., On amicable numbers. Publ. Math. Debrecen (1955), 108-111.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[Po15] Pomerance, Carl, On amicable numbers. (2015), 321-327.
[Po81] Pomerance, Carl, On the distribution of amicable numbers. II. J. Reine Angew. Math. (1981), 183-188.

Imports (1)

FormalConjecturesUtil

Imported by 2 files

abbrev Erdos830.A Let $A(x)$ count the number of amicable pairs $1\leq a\leq b\leq x$ line 43 source
Source documentation
Let $A(x)$ count the number of amicable pairs $1\leq a\leq b\leq x$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Finset.card, Prod, Nat, Finset.filter, Erdos830.A.match_1, And, LE.le, IsAmicable, Prod.fst, Prod.snd, Classical.propDecidable, SProd.sprod, Finset, Finset.Icc, OfNat.ofNat, Nat.floor
15 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev A (x : ℝ) : ℝ :=
…
```
theorem Erdos830.erdos_830.parts.i Erdos Problem 830, Part 1 We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$ sorry line 51 source
Source documentation
Erdos Problem 830, Part 1 We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. Are there infinitely many amicable pairs?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Prod, Nat, Set.ofPred, Erdos830.erdos_830.parts.i.match_1, IsAmicable
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_830.parts.i : answer(sorry) ↔ {(a, b) | IsAmicable a b}.Infinite := by
…
```
Mentions: IsAmicable
theorem Erdos830.erdos_830.parts.ii Erdos Problem 830, Part 2 We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$ sorry line 60 source
Source documentation
Erdos Problem 830, Part 2 We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then is it true that $$A(x) > x^{1-o(1)}?$$

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, Filter.Eventually, LT.lt, HPow.hPow, HSub.hSub, Erdos830.A
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_830.parts.ii : answer(sorry) ↔ ∃ o : ℝ → ℝ, o =o[atTop] (1 : ℝ → ℝ) ∧ ∀ᶠ x in atTop,
    x ^ (1 - o x) < A x := by
…
```
Mentions: Erdos830.A
theorem Erdos830.erdos_830.variants.erdos We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$ sorry line 69 source
Source documentation
We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then one can show that $A(x) = o(x)$.

depends on sorry what this means
Statement uses: Asymptotics.IsLittleO, Real, Filter.atTop, Erdos830.A, id
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_830.variants.erdos : A =o[atTop] id := by
…
```
Mentions: Erdos830.A
theorem Erdos830.erdos_830.variants.pomerance We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$ sorry line 78 source
Source documentation
We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then one can show that $A(x) \leq x \exp(-(\log x)^{1/3})$.

depends on sorry what this means
Statement uses: Filter.Eventually, Real, LE.le, Erdos830.A, HMul.hMul, Real.exp, Neg.neg, Real.nthRoot, OfNat.ofNat, Nat, Real.log, Filter.atTop
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_830.variants.pomerance : ∀ᶠ x in atTop, A x ≤ x * rexp (- Real.nthRoot 3 x.log) := by
…
```
Mentions: Erdos830.A, Real.nthRoot
theorem Erdos830.erdos_830.variants.pomerance_stronger We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$ sorry line 88 source
Source documentation
We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then one can show that $A(x) \leq x \exp(-(\tfrac{1}{2}+o(1))(\log x\log\log\log x)^{1/2})$. (erdosproblems.com has $\log\log x$; [Po15, Theorem 1.1] has $\log\log\log x$.)

depends on sorry what this means
Statement uses: Exists, Real, And, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, Filter.Eventually, LE.le, Erdos830.A, HMul.hMul, Real.exp, Neg.neg, HAdd.hAdd, HDiv.hDiv, Nat, Real.sqrt, Real.log
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_830.variants.pomerance_stronger :
    ∃ o : ℝ → ℝ, o =o[atTop] (1 : ℝ → ℝ) ∧
    ∀ᶠ x in atTop, A x ≤ x * rexp (- (1/ 2 + o x) * √(x.log * x.log.log.log)) := by
…
```
Mentions: Erdos830.A