Meaning of the expression $p^\alpha \mid \mid n$

Question

Because I cannot find it from the textbook (maybe too much?..). By the way, when I am revising arithmetic function, I saw a new symbol, related to divisibility.

For $d | n$, it means $d$ is divisible by $n$. That's easy, which learned in the first chapter.

However, what I concern is I find something $p^{\alpha}||n$ ?! I don't know what does it mean... Can anyone just help me with it? Thanks

Answer 1

Typically, $p^a \mid \mid n$ means that $p^a \mid n$, but $p^{a+1} \nmid n$. In words, this means that $p^a$ is the largest power of $p$ dividing $n$. In other notations, this is sometimes written $\mathrm{ord}_p(n) = a$.

Answer 2

$p^a\|n$ means that $p^a\mid n$ but $p^{a+1}\nmid n$.

Some authors use $m\|n$ to mean that $m\mid n$ and $\gcd(m,n/m)=1$.

Answer 3

You may read it as "precisely divides" or as a short for $p^\alpha\mid n\land p^{\alpha+1}\nmid n$.

Note that we need the exponential on the left, you can't really say $x\|y$ with arbitrary expression forms for $x$.