[seqfan] Re: -5 is not a prime

[jean-paul allouche]

So that "the" solution could be to reserve strictly (in the OEIS) the 
word "prime"
to the usual primes : 2, 3, 5, 7, ... and to specify "prime in Z" or 
"Z-prime",
resp. "Z[i]-prime" etc. for variations on the definition to sets larger 
than the
positive integers. Acceptable?

jean-paul



Le 04/01/2021 à 04:13, William Keith a écrit :
> 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.
>
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