[seqfan] Re: -5 is not a prime

[Neil Sloane]

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.

Then he gives a list of the primes less than 100, and in the third sentence
he states the Fundamentalsatz der Zahlentheorie
about unique factorization.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sun, Jan 3, 2021 at 6:12 PM Jim Nastos <nastos at gmail.com> wrote:

> Ali,
>   Yes, you can appeal to the 'natural ordering' of primes when you restrict
> your view to positive integers. But you have to generalize concepts when
> generalizing your set of "numbers". Considering all integers (Z), you
> suddenly lose the property of having a smallest element, for instance. Many
> people would say that the well-ordering principle is a cornerstone of
> understanding as well, but this only really applies to N.
>   Embracing the idea that 1,-1 are units for Z (and 1,-1,i,-i for Z[i]) is
> the cleanest way to generalize primality outside of N.
> Jim
>
> On Sun, Jan 3, 2021 at 12:36 PM Ali Sada via SeqFan <seqfan at list.seqfan.eu
> >
> wrote:
>
> >   Any change in the order of the prime factors would change the "natural
> > numbers" drastically.
> > Here is the positive integers sequence based on prime factors:a(1) =
> 1a(2)
> > = p(1)a(3) = p(2)a(4) = p(1) * p(1)a(5) = p(3)a(6)= p(1) * p(2)etc.
> >
> >  Please take a look at A064989 (if 1 were the first prime number) and
> > A331025 (if 2 were not a prime factor). What would happen to the "natural
> > order" of numbers if we consider -5 a prime?
> >
> >  I think unique factorization is a cornerstone in our understanding of
> > natural numbers.
> >
> > Happy New Year,
> >
> > Ali
> >
> >
> >
> >
> >
> >
> >
> >
> >     On Sunday, January 3, 2021, 1:18:29 PM EST, Neil Sloane <
> > njasloane at gmail.com> wrote:
> >
> >  Conway's use of -1 as a prime was just a convention that made certain
> > calculations easier.  But it should not be mentioned in public.  -1 is a
> > unit, not a prime.
> >
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, President, OEIS Foundation.
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > Email: njasloane at gmail.com
> >
> >
> >
> > On Sun, Jan 3, 2021 at 1:04 PM jean-paul allouche <
> > jean-paul.allouche at imj-prg.fr> wrote:
> >
> > > Dear Neil, dear all
> > >
> > > The convention that -1 might be considered as a prime has been used,
> > > e.g., by Conway in his book "The Sensual (Quadratic) Form". On Page 95
> > > for example he writes "Note that we treat -1 as a prime just like the
> > > others...".
> > > On Page 104, he writes "This includes the case p=-1 if by "prime to -1"
> > we
> > > mean "positive"". This is also used to define Kronecker symbols
> > > generalizing
> > > Legendre symbols. In other words, it makes several definitions etc.
> > > "simpler",
> > > but should certainly not be used without caution, as you say about
> unique
> > > factorizations.
> > >
> > > Best wishes
> > > Happy New Year!
> > > jean-paul
> > >
> > >
> > >
> > > Le 03/01/2021 à 18:02, Neil Sloane a écrit :
> > > > Amiram Eldar tells me that Mathematica considers -1 times a prime to
> > be a
> > > > prime.
> > > >
> > > > Select[Range[-10, 10], PrimeQ]
> > > > {-7, -5, -3, -2, 2, 3, 5, 7}
> > > >
> > > > This is wrong, and dangerous (it has led people to make mistakes in
> > > > sequences).
> > > >
> > > > If negative numbers were primes we would lose unique factorization, a
> > > > cornerstone of mathematics.
> > > >
> > > > --
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