一道数论题目

yoyo987654

请教思路

hbghlyj

(a−b)(a−c)(a−d)(b−c)(b−d)(c−d) divisible by 12
For a,b,c,d∈N prove that (a−b)(c−a)(d−a)(c−b)(d−b)(d−c) is divisible by 12
(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)  is divisible by 12

Czhang271828

这里写过显然法. 考虑公差是 $1$ 的等差数列即可.

hbghlyj

根据楼上写出：

1. Det[Table[Binomial[x[i], j], {i, 0, 3}, {j, 0, 3}]] // Factor

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\[\begin{vmatrix} \binom{x_0}0 & \binom{x_0}1 & \binom{x_0}2 & \binom{x_0}3 \\
\binom{x_1}0 & \binom{x_1}1 & \binom{x_1}2 & \binom{x_1}3 \\
\binom{x_2}0 & \binom{x_2}1 & \binom{x_2}2 & \binom{x_2}3 \\
\binom{x_3}0 & \binom{x_3}1 & \binom{x_3}2 & \binom{x_3}3 \end{vmatrix}
=\frac{1}{12} \left(x_0-x_1\right) \left(x_0-x_2\right) \left(x_0-x_3\right) \left(x_1-x_2\right) \left(x_1-x_3\right) \left(x_2-x_3\right)
\]

yoyo987654

hbghlyj 发表于 2023-9-1 16:19
(a−b)(a−c)(a−d)(b−c)(b−d)(c−d) divisible by 12
For a,b,c,d∈N prove that (a−b)(c−a)(d−a)(c ...

理解了，我开始时算错，计得是6，考虑少了情况

hbghlyj

整数值单变量多项式的空间有一个基 $\{\binom x0,\binom x2,\binom x3,\dots\}$ 我想问一下，这个可以推广到多变量多项式吗？