Is it true that, for every $r$, there is a $k$ such that
if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers,
all of length at least $k$, then
$$
\prod_{1\leq i\leq r}\prod_{m\in I_i}m
$$
is not a perfect power?
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theorem erdos_930 :
answer(sorry) ↔ ∀ r > 0, ∃ k, ∀ I₁ I₂ : Fin r → ℕ,
(∀ i : Fin r, 0 < I₁ i ∧ I₁ i + k ≤ I₂ i + 1) →
(∀ i j : Fin r, i < j → I₂ i < I₁ j) →
¬ IsPower (∏ i : Fin r, ∏ m ∈ Icc (I₁ i) (I₂ i), m)
