What's a good way to write "x does not divide y"?

An alternative to \nmid is to use the \centernot command from the centernot package. The resulting \centernot\mid symbol aligns perfectly with \mid and has a more pronounced slash than \nmid:

(On the right, the image shows how the commands behave in sub/superscript.)

$x\nmid y$ saves the day.

Another good looking (best to me) and easy option is to use the command \notdivides from the mathabx package. The code

\documentclass{article}
\usepackage{mathabx}
\begin{document}
\[ \prod_{a \notdivides b}^{a \notdivides b} a \notdivides b \]
\end{document}

creates the output

The negating line is longer than \nmid's but shorter than \centernot's.