What is the symbol for primes?

Question

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?

Answer 1

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).

Answer 2

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.

Answer 3

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 $\mathrm{P}$.

Answer 4

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.

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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).