Erdős Problem 1108

Reference: erdosproblems.com/1108

def Erdos1108.FactorialSums : Set ℕ

The set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials.

Equations

• Erdos1108.FactorialSums = {m : ℕ | ∃ (S : Finset ℕ), m = ∑ n ∈ S, n.factorial}

def Erdos1108.IsPowerful (n : ℕ) : Prop

A number is powerful if each prime factor appears with exponent at least 2.

Equations

• Erdos1108.IsPowerful n = ∀ (p : ℕ), Nat.Prime p → p ∣ n → p ^ 2 ∣ n

theorem Erdos1108.erdos_1108.k_th_powers : sorry ↔ ∀ k ≥ 2, {a : ℕ | a ∈ FactorialSums ∧ ∃ (m : ℕ), m ^ k = a}.Finite

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?

theorem Erdos1108.erdos_1108.powerful_numbers : sorry ↔ {a : ℕ | a ∈ FactorialSums ∧ IsPowerful a}.Finite

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?