[seqfan] Re: -5 is not a prime

[Peter Luschny]

The terminological question is easy to clarify. The principal
ideal domains form the conceptual framework of the discourse.
Here we can define prime elements.

ZZ is a principal ideal domain and here, the positive prime
elements are the prime numbers.

So far, everything is simple and systematic. The only question
left is how prime elements are defined in ZZ. Let us consider
this approach:

n is a prime number  in ZZ <=> n has exactly two divisors,
                               which are {1, n}.
n is a prime element in ZZ <=> n has exactly four divisors,
                               which are {−n, −1, 1, n}.

However, these formulations are horrible. What do I have to
expect next, an algebraic structure where a prime has exactly
six divisors?

My criticism of A000040 is that it makes precisely that definition.
That is very unsatisfactory and more of a caricature of a concept.
And this for a nice, core sequence!

In order not to appear as unconstructive, I add how I think
it should read:

p in ZZ is a prime element <=>
   (1) p not in {-1, 0, 1};
   (2) For all a, b in ZZ \ {0}
       p | a*b => p | a or p | b.

This definition is longer, but whoever doesn't bother to understand
exactly this, will never know what a prime number is.

Peter