TITLE: Generalisations of primes
QUESTION [5 upvotes]: I've read of (normal) primes, Gaussian primes and Eisenstein primes, which all uses different ways to define an integer to be a prime. For instance, $2$ factors into $1-i$ and $1+i$ for guassian primes. Although I'm thinking this might only be the surface, and I was wondering: Are there any generalisations of primes of special forms, or are Gaussian and Eisenstein primes the only interesting ways to define such primes?

REPLY [2 votes]: Perhaps not what you had in mind but I think the generalization of the prime number theorem and the corresponding generalization of primes into numbers with greater than one factor (sometimes called k-primes) qualifies as an interesting generalization. 
The generalized PNT, in which $\pi_k(n)$ is the number of k-primes not exceeding n, is:
$$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln \ln n)^{k-1}}{(k-1)!}.$$