PerfectNumbers.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

5 declarations, 5 contain sorry, 0 declared as axioms

Perfect numbers

A perfect number is a positive integer that equals the sum of its proper divisors (i.e., all its positive divisors excluding the number itself).

For example, 6 is perfect because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. Similarly, 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28.

All known perfect numbers are even. Several open problems about perfect numbers are formalised here:

Are there infinitely many perfect numbers?
Are there infinitely many even perfect numbers?
Do odd perfect numbers exist?

References:

Imports (1)

FormalConjecturesUtil

theorem PerfectNumbers.infinitely_many_perfect Infinitely many perfect numbers conjecture sorry line 55 source
Source documentation
Infinitely many perfect numbers conjecture. Are there infinitely many perfect numbers?
Reference: Wikipedia

L4DEL explanation
Infinitely many perfect numbers conjecture. Are there infinitely many perfect numbers? Reference: Wikipedia
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, Nat.Perfect
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 infinitely_many_perfect :
    answer(sorry) ↔ {n : ℕ | Perfect n}.Infinite := by
…
```
theorem PerfectNumbers.infinitely_many_even_perfect Infinitely many even perfect numbers conjecture sorry line 71 source
Source documentation
Infinitely many even perfect numbers conjecture. Are there infinitely many even perfect numbers?
This is equivalent to asking whether there are infinitely many Mersenne primes, since by the Euclid–Euler theorem an even number is perfect if and only if it has the form $2^{p-1}(2^p - 1)$ where $2^p - 1$ is a Mersenne prime.
Reference: Wikipedia

L4DEL explanation
Infinitely many even perfect numbers conjecture. Are there infinitely many even perfect numbers? This is equivalent to asking whether there are infinitely many Mersenne primes, since by the Euclid–Euler theorem an even number is perfect if and only if it has the form $2^{p-1}(2^p - 1)$ where $2^p - 1$ is a Mersenne prime. Reference: Wikipedia
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, And, Nat.Perfect, Even
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinitely_many_even_perfect :
    answer(sorry) ↔ {n : ℕ | Perfect n ∧ Even n}.Infinite := by
…
```
theorem PerfectNumbers.odd_perfect_number_conjecture Odd Perfect Number Conjecture sorry line 83 source
Source documentation
Odd Perfect Number Conjecture. The Odd Perfect Number Conjecture states that all perfect numbers are even.
Reference: Wikipedia

L4DEL explanation
Odd Perfect Number Conjecture. The Odd Perfect Number Conjecture states that all perfect numbers are even. Reference: Wikipedia
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, Nat.Perfect, Even
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem odd_perfect_number_conjecture (n : ℕ) (hn : Perfect n) : Even n := by
…
```
theorem PerfectNumbers.odd_perfect_number.lower_bound A known result: If an odd perfect number exists, it must be greater than $10^{1500}$ and must have at least 101 prime factors (including multiplicities) sorry line 94 source
Source documentation
A known result: If an odd perfect number exists, it must be greater than $10^{1500}$ and must have at least 101 prime factors (including multiplicities).
Reference: Pascal Ochem, Michaël Rao (2012). "Odd perfect numbers are greater than 10^1500"

depends on sorry what this means
Statement uses: Nat, Odd, Nat.Perfect, And, GT.gt, HPow.hPow, OfNat.ofNat, GE.ge, List.length, Nat.primeFactorsList
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem odd_perfect_number.lower_bound (n : ℕ) (hn : Odd n) (hp : Perfect n) :
    n > 10^1500 ∧ (n.primeFactorsList).length ≥ 101 := by
…
```
theorem PerfectNumbers.odd_perfect_number.euler_form A known result: If an odd perfect number exists, it must be of the form $p^α * m^2$ where $p$ is prime, $p \equiv 1 \pmod{4}$, $\alpha \equiv 1 \pmod{4}$, and $p \nmid m$ sorry line 108 source
Source documentation
A known result: If an odd perfect number exists, it must be of the form $p^α * m^2$ where $p$ is prime, $p \equiv 1 \pmod{4}$, $\alpha \equiv 1 \pmod{4}$, and $p \nmid m$.
Reference: Euler's theorem on odd perfect numbers.
Formal proof linked here provided by AlphaProof.

depends on sorry what this means
Statement uses: Nat, Odd, Nat.Perfect, Exists, And, Nat.Prime, Int.ModEq, OfNat.ofNat, Int, Not, Dvd.dvd, Eq, HMul.hMul, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem odd_perfect_number.euler_form (n : ℕ) (hn : Odd n) (hp : Perfect n) :
    ∃ (p m α : ℕ),
      p.Prime ∧
      p ≡ 1 [ZMOD 4] ∧
      α ≡ 1 [ZMOD 4] ∧
      ¬ p ∣ m ∧
      n = p^α * m^2 := by
…
```