Let $x_1,...,x_n$ be distinct integers. Prove that
$\prod_{i
I know there is a solution using determinant of a matrix, but I can't remember it now. Any help will be appreciated.
Let $x_1,...,x_n$ be distinct integers. Prove that
$\prod_{i
I know there is a solution using determinant of a matrix, but I can't remember it now. Any help will be appreciated.
If $x_1$, $\ldots$, $x_n$ are integers, then $\prod_{1\le i
You can see this by starting with the formula for the determinant of the Vandermonde matrix, $\det\left( (x_i^{j-1})_{i,j=1,\ldots,n}\right) = \prod_{1\le i
Here's a very direct approach. For each prime up to $n-1$, we just need to check that $ \text{val}_p\left( \prod_{i
To do this, notice that setting $x_i=i$ gives the minimum possible number of pairs $(i,j)$ such that $x_i-x_j$ is divisible by $p$ (or divisible by $p^2$, or by $p^3$, or any $p^k
Thus, $\displaystyle \prod_{i