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Given integers $x_1, x_2, \dotsc, x_n$, prove that the expression $$ \prod \limits_{1\leq i<j\leq n}\frac{x_i - x_j}{i-j} $$ is always an integer.

I think induction should work, but I couldn't get anywhere with it.

Mike Pierce
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John
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  • This is false. For $x_1 = x_2 = 1$ and $x_3 = 2$ the sum is $3/2$. – vonbrand Sep 04 '15 at 19:24
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    @vonbrand: No, if $x_i$ are not distinct, the product is trivially 0. – Alex R. Sep 04 '15 at 19:24
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    @AlexR., brain malfunction. Saw $\sum$. – vonbrand Sep 04 '15 at 19:33
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    A slight trick is needed to reduce this problem to the earlier one, which concerns on the face positive integers. However the trick is the obvious one: since only differences $x_i - x_j$ are present, the expression remains the same when a large positive constant is added to all the variables. – hardmath Sep 04 '15 at 22:15
  • Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. – Fly by Night Sep 04 '15 at 23:10

1 Answers1

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I don't know of a really easy of proving this fact, but here is one possible proof. The product $\prod_{1\leq i<j\leq n}(i-j)=\prod_{i=1}^ni!$ is a super factorial. The idea is that $\prod_{1\leq i<j\leq n}(x_i-x_j)$ is a vandermonde determinant, i.e. a determinate of the matrix:

$$\begin{pmatrix} 1 & x_1 & x_1^2\cdots & x_1^{n-1}\\ 1 & x_2 & x_2^2\cdots & x_2^{n-1}\\ \vdots & \vdots & \vdots & \vdots\\ 1 & x_n & x_n^2\cdots & x_n^{n-1} \end{pmatrix}.$$

The key is that by applying column operations to the above matrix, we can transform it the determinant of:

$$\begin{pmatrix} 1 & p_1(x_1) & p_2(x_1)\cdots & p_{n-1}(x_1)\\ 1& p_1(x_2) & p_2(x_2)\cdots & p_{n-1}(x_2)\\ \vdots & \vdots & \vdots & \vdots\\ 1 & p_1(x_n) & p_2(x_n)\cdots & p_{n-1}(x_n) \end{pmatrix},$$

where $p_i(x)$ is any collection of polynomials that are monic (having leading coefficient 1). In this case you can choose $p_i(x)=x(x-1)\cdots (x-i+1)=\binom{x}{i}i!$. Thus the product $\prod_{i=1}^ni!$ factors out and you're left with a determinant of binomial coefficients, which is an integer.

Alex R.
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