# What is the standard notation to represent the set of primes?

I have seen $\mathbb Z_p$. Are there others, perhaps $\mathbb N_p$?

or the set of natural numbers where the totient of $n$ equal $n - 1$ ? $$\{n \in \mathbb N \mid n\ge2,\phi (n) = n-1\}$$

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I have seen $\mathbb N_p$ –  Eli Elizirov Jul 31 '13 at 13:16
Do you need a notation? –  lhf Jul 31 '13 at 13:34
Well, "Let $p\in\{n\in\mathbb N\mid \phi(n)=n-1\}$" is only applicable, if you really want to to confuse the reader. But then, I would prefer: "Let $n$ be an integer, such that $n\mid (n-1)!+1$". –  Tomas Jul 31 '13 at 13:35
@Tomas: that makes $1$ prime, unfortunately. –  Chris Eagle Jul 31 '13 at 18:11
@ChrisEagle: Oops, then it's an obfuscation of "Let $n\in\mathbb P\cup\{1\}$". :P –  Tomas Jul 31 '13 at 20:10

## 5 Answers

First: $\mathbb{Z}_p$ does not refer to the set of primes. Depending on the context, it either refers to $\mathbb{Z}/p\mathbb{Z}$ or the $p$-adic integers.

The most common notation that I have seen for the set of primes is $\mathbb{P}$; however, it is not universally used, and so you should make sure to define it whenever you use it.

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$\mathbb{P}$ is usually projective space, as in $\mathbb{RP}^n$ or $\mathbb{CP}^n$ –  Eric Jablow Jul 31 '13 at 13:57

$\mathbb{P}$ seems to be fairly common.

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It is surprisingly common to eschew any such notation and, instead, rely on a convention that $p$ (and $p'$, $p_n$ etc) always denotes a prime number.

In fact, you can see that convention used in a recent question right here.

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I have seen $\mathbb N_p$ and also $\mathbb P$.

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Is this the case where P = NP? –  András Hummer Jul 31 '13 at 15:12

IF we use $\mathbb P$, we have to clarify that it shall be the set of Primes like this: $\mathbb P = \{ p \in \mathbb N \; | \; \text{p is prime}\}$. As far as I know there is no official definition for the set of primes. Mostly seen in my literature is e.g. Let a $\in \mathbb C$, let p be prime.

Source: I'm studying math @ Free University of Berlin.

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What does "official" mean here? –  Andres Caicedo Aug 1 '13 at 1:53
We learned, that the natural numbers are defined as $\mathbb N = ...$ so I'd consider this a "official" definition, since somehow most mathematicians agreed to this. –  swiknaba Aug 1 '13 at 10:14