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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$\begingroup$ I have seen $\mathbb N_p$ $\endgroup$– Eli ElizirovJul 31, 2013 at 13:16
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$\begingroup$ Do you need a notation? $\endgroup$– lhfJul 31, 2013 at 13:34
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4$\begingroup$ 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 $n\mid (n-1)!+1$". $\endgroup$– TomasJul 31, 2013 at 13:35
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$\begingroup$ @Tomas: that makes $1$ prime, unfortunately. $\endgroup$– Chris EagleJul 31, 2013 at 18:11
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$\begingroup$ @ChrisEagle: Oops, then it's an obfuscation of "Let $n\in\mathbb P\cup\{1\}$". :P $\endgroup$– TomasJul 31, 2013 at 20:10
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5 Answers
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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.
answered Jul 31, 2013 at 13:15
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7$\begingroup$ $\mathbb{P}$ is usually projective space, as in $\mathbb{RP}^n$ or $\mathbb{CP}^n$ $\endgroup$ Jul 31, 2013 at 13:57
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$\mathbb{P}$ seems to be fairly common.
answered Jul 31, 2013 at 13:15
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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.
answered Jul 31, 2013 at 13:20
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I have seen $ \mathbb N_p $ and also $ \mathbb P$.
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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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$\begingroup$ 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. $\endgroup$– swiknabaAug 1, 2013 at 10:14