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

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

• It means the highest exponent of $p$ that divides $n$ is $\alpha$. So $p^{\alpha} \mid n$ but $p^{\alpha+1} \nmid n$. Also $d \mid n$ means $d$ divides $n$ and not $d$ is divisible by $n$. – Anurag A Nov 23 '18 at 7:22

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 $$\mathrm{ord}_p(n) = a$$.
$$p^a\|n$$ means that $$p^a\mid n$$ but $$p^{a+1}\nmid n$$.
Some authors use $$m\|n$$ to mean that $$m\mid n$$ and $$\gcd(m,n/m)=1$$.
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 $$x$$.