$\prod_{1\leq i<j\leq n}\dfrac{x_i-x_j}{i-j}\in\mathbb Z$

Czhang271828

对整数 $\{x_i\}_{1\leq i\leq n}$, 证明 $\prod_{1\leq i<j\leq n}\dfrac{x_i-x_j}{i-j}$ 是整数.

证明

由于 $\det(\binom{x_i}{j})\in \mathbb Z$, 根据 Vandermonde 行列式, 显然.

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