What does double vertical lines $\|$ mean in number theory?

Question

I'm reading the 2002 paper P is in PRIMES and I stumbled upon this symbol $\|$ in the chapter 2:

Suppose $n$ is composite. Consider a prime $q$ that is a factor of $n$ and let $q^k\ ||\ n$. Then $q^k$ does not divide $n \choose q$ and is coprime to $a^{n−q}$ and hence the coefficient of $X^q$ is not zero $(\mod n\ )$. Thus $((X + a)^n − (X^n + a))$ is not identically zero over $Z^n$.

I am very new to number theory and it is not mentioned anywhere in the paper what $\|$ denotes. I tried looking it up in the Wikipedia List of mathematical symbols but I couldn't find anything.

Does anyone know what $\|$ means in this context?

|| is right below | (and $\not\mid$) in the Wikipedia list; you can get to it by searching the page for "exact divisibility" (or just "divisibility"). — Barry Cipra, Oct 21 '15 at 18:08
@BarryCipra Ooh cool! Thanks! I tried searching for the ascii || but couldn't find it. Looks it it uses some utf-8 symbols for the that (namely ∣∣). — Archy Will He 何魏奇, Oct 21 '15 at 18:24

score 9 · Accepted Answer · answered Oct 21 '15 at 17:56

9

It means that $q^k$ fully divides $n,$ that is, $q^k$ divides $n$ but $q^{k+1}\nmid n.$

answered Oct 21 '15 at 17:56

CIJ

3,437
1
12
19

Thanks. But what does $\nmid$ mean? Not dividable? But wouldn't that be true iff $q$ is not dividable by $n$. – Archy Will He 何魏奇 Oct 21 '15 at 18:02
Yes, $a\nmid b$ means that $b$ is not divisible by $a.$ – CIJ Oct 21 '15 at 18:03
Thank you! So just for confirmation, $$a^n | b \Rightarrow a^n = b (\ mod\ 0\ )$$ or $$(a^n | b \Rightarrow a^n = b (\ mod\ 0\ )) \land \neg (a = b (\ mod\ 0\ ))$$? From what you said it sounded like the later. But it feels like you are referring to the former. – Archy Will He 何魏奇 Oct 21 '15 at 18:07
1

Oops I meant to but $a^n$ inside mod and leave 0 outside. I got it now, thank you so much! – Archy Will He 何魏奇 Oct 21 '15 at 18:14
4

Note: $a^n \vert\vert b$ is defined only when $a$ is prime. – GEdgar Oct 21 '15 at 18:15
And my parenthesis was wrong after the 6th edit .. oops. – Archy Will He 何魏奇 Oct 21 '15 at 18:16
@GEdgar Ooh. So more formally we have $$\forall n \in \mathbb{N}, \forall b \in \mathbb{Z}, \forall a \in \text{PRIMES} \ a^n | b $$ ? – Archy Will He 何魏奇 Oct 21 '15 at 18:19

What does double vertical lines $\|$ mean in number theory?

1 Answers1

Linked