[seqfan] Re: -5 is not a prime

[Alonso Del Arte]

> If negative numbers were primes we would lose unique factorization, a
cornerstone of mathematics.

With all due respect,unique factorization is not a cornerstone of
mathematics, it's merely a property of our favorite number domain, and
perhaps infinitely many others, but not all others. And in any case, since
ordering doesn't bother us, why should multiplication by units?

−5 is not prime in *ℤ*[*i*], but then neither is 5, 5*i* nor −5*i*. We see
for example that 5 = (2  − *i*)(2 + *i*). And also (1  − 2*i*)(1 + 2*i*) =
5. But that doesn't negate the fact that *ℤ*[*i*] is UFD. If there are
intelligent electricity-based life-forms, they might be puzzled that we
consider 1 to be such a canonical unit rather than  −1 or *i*.

When the symmetry of the primes multiplied by units is inconvenient for the
temporary purpose at hand, we can simply ignore it or assume it to be
implicitly understood.
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On Sun, Jan 3, 2021 at 12:02 PM Neil Sloane <njasloane at gmail.com> wrote:

> 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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Alonso del Arte
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<https://www.smashwords.com/profile/view/AlonsoDelarte>
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