FormalConjectures.ErdosProblems.«677»

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import FormalConjecturesUtil

Erdős Problem 677

Reference: erdosproblems.com/677


namespace Erdos677


open Finset

Erdős expected very few solutions for $M(n, k) = M(m, l)$, where $m \geq n + k$ and $l > 1$. The only solutions he knew were the following.

@[category test, AMS 11]
lemma lcmInterval_eq_example1 : lcmInterval 4 3 = lcmInterval 13 2 ∧
                                lcmInterval 3 4 = lcmInterval 19 2 := by⊢ lcmInterval 4 3 = lcmInterval 13 2 ∧ lcmInterval 3 4 = lcmInterval 19 2 decideAll goals completed! 🐙

Denote by $M(n, k)$ the least common multiple of the finite set ${n+1, \dotsc, n+k}$. Is it true that for all $m \geq n + k$, we get $M(m, k) \neq M(n, k)$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_677 :
    ∀ (m n k : ℕ), k > 0 → m ≥ n + k → lcmInterval m k ≠ lcmInterval n k := by⊢ ∀ (m n k : ℕ), k > 0 → m ≥ n + k → lcmInterval m k ≠ lcmInterval n k
  sorryAll goals completed! 🐙


-- TODO: Add the other statements from the reference.