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import FormalConjecturesUtil
open ArithmeticFunction.sigma
namespace Erdos412
Let $σ_1(n)=σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$. Is it true that, for every $m, n ≥ 2$, there exist some $i, j$ such that $σ_i(m) = σ_j(n)$?
@[category research open, AMS 11]
theorem erdos_412 : answer(sorry) ↔ ∀ᵉ (m ≥ 2) (n ≥ 2), ∃ i j, (σ 1)^[i] m = (σ 1)^[j] n := ⊢ True ↔ ∀ m ≥ 2, ∀ n ≥ 2, ∃ i j, (⇑(σ 1))^[i] m = (⇑(σ 1))^[j] n
All goals completed! 🐙
end Erdos412