469.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 469

References:

erdosproblems.com/469
[Le25] Lewis, Z. J., On the convergence of the reciprocal sum of primitive pseudoperfect numbers. Preprint (2025).

Imports (1)

FormalConjecturesUtil

def Erdos469.Nat.IsSumDivisors The proposition that n is a sum of distinct proper divisors line 34 source
Source documentation
The proposition that n is a sum of distinct proper divisors.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Finset, And, LE.le, Nat.properDivisors, Eq, Finset.sum
4 instances, coercions and auxiliary constants left out.
```
def Nat.IsSumDivisors (n : ℕ) : Prop :=
…
```
theorem Erdos469.erdos_469 Let $A$ be the set of all $n$ such that $n = d_1 + ⋯ + d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m ∣ n$ with $m < n$ sorry line 51 source
Source documentation
Let $A$ be the set of all $n$ such that $n = d_1 + ⋯ + d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m ∣ n$ with $m < n$. Does: $$ \sum_{n ∈ A} \frac 1 n $$ converge?
Yes: the sum converges. This was proved by Lewis [Le25], whose proof has been formalized in Lean.

depends on sorry what this means
Statement uses: Iff, True, Summable, Real, Nat, Set.ofPred, And, LT.lt, OfNat.ofNat, Erdos469.Nat.IsSumDivisors, Dvd.dvd, Not, HDiv.hDiv, Membership.mem, Set, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_469 :
    letI A := {n : ℕ | 0 < n ∧ n.IsSumDivisors ∧ ∀ m < n, m ∣ n → ¬ m.IsSumDivisors}
…
```