
New Number Systems Seek Their Lost Primes - merrier
https://www.quantamagazine.org/ideal-numbers-seek-their-lost-primes-20170302
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ginnungagap
_All four of those factors are prime in the new number ring, giving 6 a dual
existence_

Those factors are irreducibles, but they aren't primes, which is the reason
why the uniqueness of factorization fails.

A nonzero non unit element of a ring is called prime if p|ab implies p|a or
p|b.

A nonzero non unit element of a ring is called irreducible if p=ab implies
that a is a unit or b is a unit (invertible element).

Primes are irreducibles in an integral domain, but the converse is true in
unique factorization domains and Z[√-5] is not one.

~~~
wodenokoto
That's a statement that goes quite against the claims of the article and I
kinda want to ask for a citation ... But I already feel that this discussion
is way over my head and I wouldn't understand any journal articles that would
clear this up ...

~~~
ginnungagap
Wikipedia isn't a good source, but it uses 3 as an element of Z[√-5] as an
example so it's quite relevant:
[https://en.m.wikipedia.org/wiki/Irreducible_element](https://en.m.wikipedia.org/wiki/Irreducible_element)

For a more serious reference any abstract algebra book covering rings and UFDs
should do, for example it is in Dummit & Foote.

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ajuc
> Mathematicians call this new system a number “ring”; they can create an
> infinite variety of them, depending on the new values they choose to
> incorporate.

Isn't this a little wrong?

As far as I remember ring is any set with multiplication, negation, and
addition defined so that they satisfy a few conditions. No need for the "\+ b
* something" part. The usual integer numbers we use form a ring too, as well
as booleans.

I might be missing something, and it's irrelevant to the main subject of the
article.

~~~
gizmo686
"Number ring" has a specific meaning [0]. I am not sure if it is equivalent to
the integers adjoined with some element though.

>In fact the usual integer numbers we use form a ring too.

Depending on definitions, the integers can be viewed as the ring of even
numbers adjoined with the element 1. This does require that we do not define
rings to necessarily contain 1. In my experience this definition is is
becoming more of a historical footnote though.

There is also a natural generalization of adjoining multiple elements to the
natural numbers (which still results in a ring) In this case, the natural
numbers would just be a special case of adjoining 0 elements.

The ring of real numbers, in contrast, cannot be constructed by adjoining any
finite set of elements to the integers.

The ring of integers mod n is also a ring, but does not contain the integers
as a subring (and therefore cannot be thought of as the integers adjoined with
any set (finite or infinite) of elements).

The polynomials with coeficients mod n form a non-finite ring which does not
contain the integers.

I suspect the point that the author was attempting to make was just that Z[√5]
formed a type of structure that mathematicians are familiar with.

[0]
[http://mathworld.wolfram.com/NumberRing.html](http://mathworld.wolfram.com/NumberRing.html)

~~~
ajuc
Oh, right, that's what I missed, number ring isn't just a ring of numbers.
Thanks.

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kevinr
What a missed opportunity, not to title this article "In Search of Lost
Primes".

(inb4 all the Proust fans downmod me to oblivion)

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Modj
"Recall that every group has a zero element that, when added to any other
number, leaves that number unchanged" not so fast.

~~~
msds
Uh, up to notation, a group better have such an element - otherwise it isn't a
group!

~~~
chowells
You're right, but phrasing things in terms of addition and zero is possibly
taking some liberties with the definition of group. Even if additive groups
and general groups are isomorphic, I agree it's a bit misleading to use
additive group naming to provide a lay definition of "group".

