FormalConjectures.ErdosProblems.«1064»

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import FormalConjecturesUtil

Erdős Problem 1064

Reference: erdosproblems.com/1064

open Nat Filter Topologynamespace Erdos1064

Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$ have asymptotic density 1. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math.

@[category research solved, AMS 11]
theorem erdos_1064 : {n | φ n > φ (n - φ n)}.HasDensity 1 := by⊢ {n | φ n > φ (n - φ n)}.HasDensity 1
  sorryAll goals completed! 🐙

Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$ such that $ϕ(n)< ϕ(n - ϕ(n))$ Reference: [GLW01] Grytczuk, A. and Luca, F. and W'ojtowicz, M., A conjecture of {E}rdős concerning inequalities for the {E}uler totient function.


@[category research solved, AMS 11]
theorem erdos_1064.variants.k2 : {n | φ n < φ (n - φ n)}.Infinite := by⊢ {n | φ n < φ (n - φ n)}.Infinite
  apply Set.infinite_of_injective_forall_mem (f := fun k : ℕ => 30 * 2 ^ k)hi⊢ Function.Injective fun k ↦ 30 * 2 ^ khf⊢ ∀ (x : ℕ), 30 * 2 ^ x ∈ {n | φ n < φ (n - φ n)}
  ·hi⊢ Function.Injective fun k ↦ 30 * 2 ^ k intro a b habhia:ℕb:ℕhab:(fun k ↦ 30 * 2 ^ k) a = (fun k ↦ 30 * 2 ^ k) b⊢ a = b
    simp only at habhia:ℕb:ℕhab:30 * 2 ^ a = 30 * 2 ^ b⊢ a = b
    have : (2 : ℕ) ^ a = 2 ^ b := by⊢ {n | φ n < φ (n - φ n)}.Infinitehia:ℕb:ℕhab:30 * 2 ^ a = 30 * 2 ^ bthis:2 ^ a = 2 ^ b⊢ a = b omegahia:ℕb:ℕhab:30 * 2 ^ a = 30 * 2 ^ bthis:2 ^ a = 2 ^ b⊢ a = bhia:ℕb:ℕhab:30 * 2 ^ a = 30 * 2 ^ bthis:2 ^ a = 2 ^ b⊢ a = b
    exact Nat.pow_right_injective (le_refl 2) thisAll goals completed! 🐙
  ·hf⊢ ∀ (x : ℕ), 30 * 2 ^ x ∈ {n | φ n < φ (n - φ n)} intro khfk:ℕ⊢ 30 * 2 ^ k ∈ {n | φ n < φ (n - φ n)}
    simp only [Set.mem_ofPred_eq]hfk:ℕ⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
    have hφ : Nat.totient (30 * 2 ^ k) = 8 * 2 ^ k := by⊢ {n | φ n < φ (n - φ n)}.Infinitehfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
      have h : 30 * 2 ^ k = 2 ^ (k + 1) * 15 := by⊢ {n | φ n < φ (n - φ n)}.Infinitek:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ φ (30 * 2 ^ k) = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) ringk:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ φ (30 * 2 ^ k) = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))k:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ φ (30 * 2 ^ k) = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
      rw [h,k:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ φ (2 ^ (k + 1) * 15) = 8 * 2 ^ kk:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ k * (2 - 1) * 8 = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) Nat.totient_mul (byk:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ (2 ^ (k + 1)).Coprime 15k:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ k * (2 - 1) * 8 = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) norm_numAll goals completed! 🐙k:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ k * (2 - 1) * 8 = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))),
        Nat.totient_prime_pow Nat.prime_two (byk:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 0 < k + 1k:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ k * (2 - 1) * 8 = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) omegaAll goals completed! 🐙k:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ k * (2 - 1) * 8 = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))),
        show Nat.totient 15 = 8 from rfl,k:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ (k + 1 - 1) * (2 - 1) * 8 = 8 * 2 ^ kk:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ k * (2 - 1) * 8 = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) Nat.add_sub_cancelk:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ k * (2 - 1) * 8 = 8 * 2 ^ kk:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ k * (2 - 1) * 8 = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))]k:ℕh:30 * 2 ^ k = 2 ^ (k + 1) * 15⊢ 2 ^ k * (2 - 1) * 8 = 8 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
      ringhfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
    have hsub : 30 * 2 ^ k - Nat.totient (30 * 2 ^ k) = 22 * 2 ^ k := by⊢ {n | φ n < φ (n - φ n)}.Infinitehfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
      rw [hφk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ 30 * 2 ^ k - 8 * 2 ^ k = 22 * 2 ^ kk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ 30 * 2 ^ k - 8 * 2 ^ k = 22 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))]k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ k⊢ 30 * 2 ^ k - 8 * 2 ^ k = 22 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
      omegahfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
    have hφsub : Nat.totient (22 * 2 ^ k) = 10 * 2 ^ k := by⊢ {n | φ n < φ (n - φ n)}.Infinitehfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
      rw [show 22 * 2 ^ k = 2 ^ (k + 1) * 11 by⊢ {n | φ n < φ (n - φ n)}.Infinitek:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) ringAll goals completed! 🐙k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)),
        Nat.totient_mul (byk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ (2 ^ (k + 1)).Coprime 11k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) norm_numAll goals completed! 🐙k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))), Nat.totient_prime_pow (byk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ Nat.Prime 2k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) norm_numAll goals completed! 🐙k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))) (byk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 0 < k + 1k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) omegaAll goals completed! 🐙k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))),
        show Nat.totient 11 = 10 from rfl,k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ (k + 1 - 1) * (2 - 1) * 10 = 10 * 2 ^ kk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k)) Nat.add_sub_cancelk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ kk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))]k:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ k⊢ 2 ^ k * (2 - 1) * 10 = 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
      ringhfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (30 * 2 ^ k - φ (30 * 2 ^ k))
    rw [hsub,hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ φ (30 * 2 ^ k) < φ (22 * 2 ^ k)hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ 8 * 2 ^ k < 10 * 2 ^ k hφ,hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ 8 * 2 ^ k < φ (22 * 2 ^ k)hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ 8 * 2 ^ k < 10 * 2 ^ k hφsubhfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ 8 * 2 ^ k < 10 * 2 ^ khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ 8 * 2 ^ k < 10 * 2 ^ k]hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ 8 * 2 ^ k < 10 * 2 ^ k
    have : (0 : ℕ) < 2 ^ k := pow_pos (byk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ k⊢ 0 < 2hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ kthis:0 < 2 ^ k⊢ 8 * 2 ^ k < 10 * 2 ^ k norm_numAll goals completed! 🐙hfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ kthis:0 < 2 ^ k⊢ 8 * 2 ^ k < 10 * 2 ^ k) khfk:ℕhφ:φ (30 * 2 ^ k) = 8 * 2 ^ khsub:30 * 2 ^ k - φ (30 * 2 ^ k) = 22 * 2 ^ khφsub:φ (22 * 2 ^ k) = 10 * 2 ^ kthis:0 < 2 ^ k⊢ 8 * 2 ^ k < 10 * 2 ^ k
    omegaAll goals completed! 🐙

open Asymptotics Filter

For any function $f(n)=o(n)$, we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math. (2002), 111--130.

@[category research solved, AMS 11]
theorem erdos_1064.variants.general_function (f : ℕ → ℕ)
    (hf : (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))) :
    {n : ℕ | φ (n - φ n) + f n < φ n}.HasDensity 1 := byf:ℕ → ℕhf:(fun n ↦ ↑(f n)) =o[atTop] fun n ↦ ↑n⊢ {n | φ (n - φ n) + f n < φ n}.HasDensity 1
  sorryAll goals completed! 🐙

end Erdos1064