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Prove that there are no $n,x,y$ with $x+n\leq y$ such that \[\mathrm{lcm}(x+1,\ldots,x+n) = \mathrm{lcm}(y+1,\ldots,y+n).\]
It follows from the Thue-Siegel theorem that, for $n$ fixed, there are only finitely many potential such $x$ and $y$.