470.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

10 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 470

Reference: erdosproblems.com/470

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Wikipedia/OddWeirdNumber.lean

def Erdos470.PrimitiveWeird Primitive weird numbers are weird numbers such that no proper divisor of $n$ are weird line 33 source
Source documentation
Primitive weird numbers are weird numbers such that no proper divisor of $n$ are weird.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Nat.Weird, Membership.mem, Finset, Nat.properDivisors, Not
2 instances, coercions and auxiliary constants left out.
```
def PrimitiveWeird (n : ℕ) := n.Weird ∧ ∀ d ∈ n.properDivisors, ¬d.Weird
```
def Erdos470.AbundancyIndex The abundancy index is the sum of the divisors of $n$ divided by $n$ line 38 source
Source documentation
The abundancy index is the sum of the divisors of $n$ divided by $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: HDiv.hDiv, Finset.sum, Nat.divisors
5 instances, coercions and auxiliary constants left out.
```
def AbundancyIndex (n : ℕ) : ℚ := (∑ d ∈ n.divisors, d) / n
…
```
theorem Erdos470.erdos_470.parts.i Are there any odd weird numbers? sorry line 44 source
Source documentation
Are there any odd weird numbers?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, Nat.Weird, Odd
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 erdos_470.parts.i : answer(sorry) ↔ ∃ n : ℕ, n.Weird ∧ Odd n := by
…
```
theorem Erdos470.erdos_470.parts.ii Are there infinitely many primitive weird numbers? sorry line 51 source
Source documentation
Are there infinitely many primitive weird numbers?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Erdos470.PrimitiveWeird
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_470.parts.ii : answer(sorry) ↔ Set.Infinite PrimitiveWeird := by
…
```
Mentions: Erdos470.PrimitiveWeird
theorem Erdos470.erdos_470.variants.weird_pos_density Benkoski and Erdős BeEr74 proved that the set of weird numbers has positive density sorry line 66 source
Source documentation
Benkoski and Erdős BeEr74 proved that the set of weird numbers has positive density.
HasPosDensity is the right reading here, rather than the positive lower density that Erdős' "positive density" usually abbreviates. Their Theorem 5 is stated as "the density of weird numbers is positive", and they first establish that the density exists at all: "It is clear that the weird numbers have a density since both the abundant numbers and the pseudoperfect numbers have a density. (A weird number is abundant and not pseudoperfect.)" The work in the paper goes into showing that density is not 0.

L4DEL explanation
Benkoski and Erdős BeEr74 proved that the set of weird numbers has positive density. HasPosDensity is the right reading here, rather than the positive lower density that Erdős' "positive density" usually abbreviates. Their Theorem 5 is stated as "the density of weird numbers is positive", and they first establish that the density exists at all: "It is clear that the weird numbers have a density since both the abundant numbers and the pseudoperfect numbers have a density. (A weird number is abundant and not pseudoperfect.)" The work in the paper goes into showing that density is not 0.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Set.HasPosDensity, Nat, Set.ofPred, Nat.Weird, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_470.variants.weird_pos_density : {n : ℕ | n.Weird}.HasPosDensity := by
…
```
theorem Erdos470.erdos_470.variants.smallest_weird_eq_70 The smallest weird number is 70 line 73 source
Source documentation
The smallest weird number is 70.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: And, Nat, LT.lt, OfNat.ofNat, Not, Nat.Weird
Proof uses: And.intro, Nat.Abundant, Nat.Pseudoperfect, False, dite, LE.le, Finset.sum, Nat.properDivisors, absurd, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right and 10 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 60 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_470.variants.smallest_weird_eq_70 : (∀ n < 70, ¬n.Weird) ∧ (70).Weird := by
…
```
theorem Erdos470.erdos_470.variants.prime_gap_imp_inf_prim_weird Documented declaration: prime_gap_imp_inf_prim_weird sorry line 104 source
Source documentation
Melfi Me15 has proved that there are infinitely many primitive weird numbers, conditional on the fact that $p_{n+1} - p_n < \frac{1}{10} \sqrt{p_n}$ for all large $n$, which in turn would follow from well-known conjectures concerning prime gaps.

L4DEL explanation
Melfi Me15 has proved that there are infinitely many primitive weird numbers, conditional on the fact that $p_{n+1} - p_n < \frac{1}{10} \sqrt{p_n}$ for all large $n$, which in turn would follow from well-known conjectures concerning prime gaps.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, LT.lt, Real, primeGap, HDiv.hDiv, Real.sqrt, Nat.nth, Nat.Prime, OfNat.ofNat, Filter.atTop, Set.Infinite, Erdos470.PrimitiveWeird
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_470.variants.prime_gap_imp_inf_prim_weird :
    (∀ᶠ n in Filter.atTop, primeGap n < √ (n.nth Nat.Prime) / 10) →
      Set.Infinite PrimitiveWeird := by
…
```
Mentions: primeGap, Erdos470.PrimitiveWeird
theorem Erdos470.erdos_470.variants.odd_weird_10_pow_21 Fang Fa22 has shown there are no odd weird numbers below $10^{21}$ sorry line 113 source
Source documentation
Fang Fa22 has shown there are no odd weird numbers below $10^{21}$.

L4DEL explanation
Fang Fa22 has shown there are no odd weird numbers below $10^{21}$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, LT.lt, HPow.hPow, OfNat.ofNat, Odd, Not, Nat.Weird
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_470.variants.odd_weird_10_pow_21 : ∀ n < 10 ^ 21, Odd n → ¬n.Weird := by
…
```
theorem Erdos470.erdos_470.variants.odd_weird_prime_div Liddy and Riedl LiRi18 have shown that an odd weird number must have at least 6 prime divisors sorry line 121 source
Source documentation
Liddy and Riedl LiRi18 have shown that an odd weird number must have at least 6 prime divisors.

L4DEL explanation
Liddy and Riedl LiRi18 have shown that an odd weird number must have at least 6 prime divisors.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, Odd, Nat.Weird, LE.le, OfNat.ofNat, Set.ncard, Set.ofPred, And, Membership.mem, Finset, Nat.divisors, Nat.Prime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_470.variants.odd_weird_prime_div :
    ∀ n : ℕ, Odd n → n.Weird → 6 ≤ {m | m ∈ n.divisors ∧ m.Prime}.ncard := by
…
```
theorem Erdos470.erdos_470.variants.abundancy_index sorry line 130 source
depends on sorry what this means
Statement uses: Nat, Nat.Weird, Not, Odd, LT.lt, Rat, Erdos470.AbundancyIndex, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
7 instances, coercions and auxiliary constants left out.

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