FormalConjectures.ErdosProblems.«458»

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import FormalConjecturesUtil

Erdős Problem 458

Reference: erdosproblems.com/458


namespace Erdos458

The least common multiple of the integers in the set ${1, \dots, n}$.

def lcm_upto (n : ℕ) : ℕ :=
  (Finset.Icc 1 n).lcm id

Let $\operatorname{lcm}(1, \dots, n)$ denote the least common multiple of ${1, \dots, n}$. Let $p_k$ be the $k$-th prime. Is it true that for all $k \geq 1$, $\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_458 :
    answer(sorry) ↔ ∀ k : ℕ, lcm_upto ((k + 1).nth Prime - 1)
     < k.nth Prime * lcm_upto (k.nth Prime) := by⊢ True ↔ ∀ (k : ℕ), lcm_upto (Nat.nth Prime (k + 1) - 1) < Nat.nth Prime k * lcm_upto (Nat.nth Prime k)
    sorryAll goals completed! 🐙


end Erdos458