[seqfan] Re: -5 is not a prime

[Ali Sada]

  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.
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
>
>
> --
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