[seqfan] Re: -5 is not a prime

[jean-paul allouche]

Sure. You are of course right!
(I was trying to track the reason
why Mathematica had this strange
definition).
best
jean-paul

Le 03/01/2021 à 19:18, Neil Sloane a écrit :
> 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/
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
> --
> Seqfan Mailing list - http://list.seqfan.eu/