Let $k_1 \geq k_2 \geq 3$. Are there only finitely many $n_2\geq n_1 + k_1$ such that $$\prod_{1\leq i\leq k_1}(n_1 + i)\ \text{and}\ \prod_{1\leq j\leq k_2} (n_2 + j)$$ have the same prime factors?

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