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import FormalConjecturesUtilErdős Problem 1108
open Nat Filter BigOperators
namespace Erdos1108
The set $A = \left{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right}$ of all finite sums of distinct factorials.
def FactorialSums : Set ℕ :=
{m : ℕ | ∃ S : Finset ℕ, m = ∑ n ∈ S, n.factorial}
A number is powerful if each prime factor appears with exponent at least 2.
def IsPowerful (n : ℕ) : Prop :=
∀ p : ℕ, p.Prime → p ∣ n → p ^ 2 ∣ n
For each $k \geq 2$, does the set $A = \left{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?
@[category research open, AMS 11]
theorem erdos_1108.parts.i : answer(sorry) ↔ ∀ k ≥ 2,
Set.Finite { a | a ∈ FactorialSums ∧ ∃ m : ℕ, m ^ k = a } := ⊢ True ↔ ∀ k ≥ 2, {a | a ∈ FactorialSums ∧ ∃ m, m ^ k = a}.Finite
All goals completed! 🐙
Does the set $A = \left{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right}$ of all finite sums of distinct factorials contain only finitely many powerful numbers?
@[category research open, AMS 11]
theorem erdos_1108.parts.ii :
answer(sorry) ↔ {a ∈ FactorialSums | IsPowerful a}.Finite := ⊢ True ↔ {a | a ∈ FactorialSums ∧ IsPowerful a}.Finite
All goals completed! 🐙
end Erdos1108