Generalisations of primes

Question

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?

Answer 1

Along the lines of the examples you've cited, the generalisations of primes in $\mathbb Z$ is prime elements in rings of integers of algebraic number fields.

The whole area of algebraic number theory is concerned with how primes in $\mathbb Z$ factors in rings of integers (and more).

Gaussian integers and Eisenstein integers are useful to answer questions about which numbers are represented by certain quadratic forms ($x^2+y^2$ in the case of Gaussian integers and $x^2-xy+y^2$ in the case of Eisenstein integers).

The general case is much harder. See the book Primes of the Form $x^2+ny^2$, by David Cox, which is excellent but not for beginners.

Answer 2

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:

$$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln \ln n)^{k-1}}{(k-1)!}.$$