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import FormalConjecturesUtil(m,k)-perfect numbers
An integer n : ℤ is (m,k)-perfect if σᵐ(n) = kn where σᵐ is the mᵗʰ iterate of the
sum of divisors function.
open ArithmeticFunction.sigma
namespace Superperfect
A positive integer $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$ where $\sigma^m$ is the $m$-th iterate of $σ$.
def PerfectFor (n m k : ℕ) : Prop := 0 < n ∧ Nat.iterate (fun x => σ 1 x) m n = k * nThere does not exist a $(2,5)$-perfect number
@[category research open, AMS 11]
theorem twoFivePerfect : ¬ ∃ n, PerfectFor n 2 5 := ⊢ ¬∃ n, PerfectFor n 2 5
All goals completed! 🐙
end Superperfect