[btilly]

You are oversimplifying more than you know.

In a number theory course around 1990, I remember reading papers from the 60s which would explicitly note whether the counts of primes that they were using were starting from 1 or 2. You can define them as not notable mathematicians, but working mathematicians in number theory still had not completely standardized 50 years ago.

Now for more fun, sit down with a group of mathematicians and ask whether they consider 0 to be a natural number. :-)

(The answer you get will vary by field. But none will consider it a particularly important question.)

[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.