What is the symbol for primes? Although there isn't much difference between $\mathbb{Z},\mathbb{N},\mathbb{I}$, they are well-known, and each one gets its own distinguished symbol. Is there any reason that primes don't get their own special symbol? Or is there an already commonly used symbol for primes?
 A: Relevant / duplicate / posted on MO: A symbol to denote the set of prime numbers. 
From the thread on MO, and from what I've seen elsewhere, the symbol
$$
\Huge\mathbb{P}
$$
is sometimes used. This doesn't really seem to be all too common though (not universal anyway).
A: Whatever you go with, remember that $\mathbb{P}$ is commonly used elsewhere, so make it very clear in your writing that you're defining it to be the set of primes.  Personally, I've never seen $\mathbb{P}$ used to denote the primes, although apparently some do.
For example, although the expression $\displaystyle \sum_{p \in \mathbb{P}} \frac{1}{p} = \infty$ has an appealing brevity to it, I've grown to just bite the bullet and write $\displaystyle \sum_{p\text{ prime} } \frac{1}{p} = \infty$.  I will concede that, given their importance, the primes not having a widely agreed-upon symbol is one of the major notational shortcomings of modern mathematics.
A: In computer science (more precisely, when dealing with algorithms), the set of all primes (or, more accurately, of all representations of primes as strings in some alphabet), is generally denoted $\mathrm{PRIMES}$ or $\mathrm{P}\scriptstyle\mathrm{RIMES}$, as is usual to denote the language associated with some decision problem. See for example $\mathrm{PRIMES}$ is in $\mathrm{P}$.
A: First of all, I think there's only one place where I've seen $\mathbb{I}$ for the integers: http://mathworld.wolfram.com/Doublestruck.html I've seen $\mathbb{N}$ slightly more often, but then you get into the issue of whether $0$ is a natural number or not, so on that count I think you're better off using $\mathbb{Z}$, possibly with a $^+$ and a $\cup \{0\}$ if needed.

At the moment I can't recall where, but I have seen $\mathbb{P}$ to denote the primes, and that's what that Mathworld page says. But it also says $\mathbb{P}^n$ is $n$-dimensional real projective space. So if you want to refer to the squares of primes with $\mathbb{P}^2$, that might get problematic.
Here's another symbol I've seen for the primes: $\mathcal{P}$. I think it was in an ArXiV paper, I doubt it was in an actual book from a library. Don't use $\mathfrak{P}$, though, that's more commonly used for a prime ideal (plus it looks like a B, to boot).
But, how often do you have to refer to the positive primes of $\mathbb{Z}$ as a set? It seems to me that about the only time you need to do that is when you need to iterate some variable (usually $p$) through all the positive primes or a subset thereof. As Kaj suggests, it might be best to write $p \textrm{ prime}$ or "$p$ runs through the primes" (the latter coming in handy if you also need to specify a condition like $p \leq n$; but some authors of actual books actually stick a $p \textrm{ prime}$ under that).
