The point about the vertical bar occurring both in division and in setbuilder notation is a good one, and I expect to point this out in the abstractmath article on sets.

]]>The problem with $m|n$ and $m/n$ is that (1) both of them involve division and (2) the placement of $n$ and $n$ are reversed.

I will admit that I have taught discrete math lots of times, where this notation comes up and the Legendre symbol does not. I have never taught number theory, introductory or otherwise. So my experience is distinctly biased.

]]>One symbol, which is also used in the study of integers, and is even more confusing because of its similarity to a notation for division, is the notation for the Legendre symbol $\left(\dfrac{a}b\right)$. That uses the horizontal line and parentheses even though those already have different meanings.

Then the vertical bar can also be used for “such that”. That can make some expressions hard to parse. For example $\{a|f(a)|n\}$

(Sorry if the notation doesn’t come out right — your blog should provide a “preview” facility and say what notation is available.)

]]>Just seems like an unneeded extra hurdle when the math is hard enough.

]]>I’d recommend changing this wording: This is certainly not true

Maybe to: This is certainly not (always) true