FormalConjectures.ErdosProblems.«828»

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import FormalConjecturesUtil

Erdős Problem 828

Reference: erdosproblems.com/828


namespace Erdos828


open scoped Nat

Is it true that, for any $a \in \mathbb{Z}$, there are infinitely many $n$ such that $$\phi(n) | n + a$$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_828 : answer(sorry) ↔ ∀ a : ℤ, Set.Infinite {n : ℕ | ↑(φ n) ∣ n + a} := by⊢ True ↔ ∀ (a : ℤ), {n | ↑(φ n) ∣ ↑n + a}.Infinite
  sorryAll goals completed! 🐙

When $n > 1$, Lehmer conjectured that $\phi(n) | n - 1$ if and only if $n$ is prime.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_828.variants.lehmer_conjecture : answer(sorry) ↔ ∀ n > 1, φ n ∣ n - 1 ↔ Prime n := by⊢ True ↔ ∀ n > 1, φ n ∣ n - 1 ↔ Prime n
  sorryAll goals completed! 🐙

It is an easy exercise to show that $\phi(n) | n$ if and only if $n = 0, 1$ or $n = 2^a 3^b$ for some $a > 0$.

@[category textbook, AMS 11, formal_proof using formal_conjectures at
"https://github.com/XC0R/formal-conjectures/blob/03e00cf8d44098d0fb06e891fca30c29769df619/FormalConjectures/ErdosProblems/828.lean#L49"]
theorem declaration uses 'sorry'erdos_828.variants.phi_dvd_self_iff_pow2_pow3 {n : ℕ} :
    φ n ∣ n ↔ n ≤ 1 ∨ ∃ᵉ (a > 0) (b), n = 2 ^ a * 3 ^ b := byn:ℕ⊢ φ n ∣ n ↔ n ≤ 1 ∨ ∃ a > 0, ∃ b, n = 2 ^ a * 3 ^ b
  sorryAll goals completed! 🐙


end Erdos828