[tel]

cperciva answered this according to the modern algebraic understanding in response to the parent. 1 is definitely not prime when you more fully categorize the elements of sets by their algebraic properties: the existence of units is a less singular phenomenon in other algebraic structures.

[btilly]

Yes. More abstractly, prime numbers are those that generate prime ideals. This is a definition which generalizes to much more complex algebraic structures.

However the last vestiges of the question about what definition was most natural didn't get settled until surprisingly recently.

[tel]

Oh, sorry for being presumptuous then. I'm not aware of the recent history.