What does notation $a|b^\infty$ mean?

Question

I have definitely crossed it before, but can't find it now. What does it mean when people write $a \mid b^\infty$?

Answer 1

If we think of $~|~$ as a binary operator returning a Boolean (true or false), we can do the standard trick for interpreting $\infty$:

$$ a | b^{\infty} \;≝\; \lim_{n \rightarrow \infty} a | b^n $$

From this, we can see that $a|b^\infty$ iff all of $a$'s prime factors divide $b$.

As an example (from Yves's comment above), $256 | 12^\infty$.

Caveat: Notation does not always mean the same thing in different papers. Definitions or contextual information from the relevant source always has priority.