[seqfan] Re: -5 is not a prime

[William Keith]

On Sun, Jan 3, 2021 at 6:47 PM Neil Sloane <njasloane at gmail.com> wrote:

> Landau's great classic book Primzahlen ("Handbuch der Lehre von der
> Verteilung der Primzahlen", 1909) is 1000 pages long, and begins on page 1
> with the sentence
>
> Unter einer Primzahl versteht man eine positive ganze Zahl, welche von 1
> verschieden und nur durch 1 und durch sich
> selbst teilbar ist.
>

And in the semiring of whole numbers that is a perfectly correct
definition.  If we admit -5 to exist and ask whether it is prime, however,
the question is being asked in -- most likely -- the integers, in which -5
is divisible by only itself, along with the units and its associates as
every number is, and hence is a prime.

Whether a number is prime depends on the setting in which one asks the
question.  Is 2 prime?  In the whole numbers, yes.  In the integers, yes.
In the Gaussian integers, no.

In the integers Z, -5 is prime.  Primality is an algebraic definition which
it fits.  In the whole numbers, -5 doesn't exist and the question is
meaningless.