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Old 2020-10-29, 07:06   #1
jwaltos
 
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Apr 2012

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Default J.H.Conway -1 is prime

Years ago I had some results which pointed to -1 as a prime number. This was done in a logically consistent manner and the only logical break would be to consider it otherwise. At the time I researched this question and had copied the contents of the newsgroup discussion which this (now broken) link pointed to:

"Date: 10/21/2000 at 03:33:50
From: Doctor Floor
Subject: Re: Is -1 Prime?

Dear Tom,

I read the discussion you had with Dr. Peterson. Indeed it is true
that in the usual definition of a prime, the prime is supposed to be a
positive integer, or natural number.

However, there are people, and not the least, who do consider -1 to be
a prime. Professor John Conway of Princeton is one of them. Although I
don't suppose it is desirable to have this in high school textbooks, I
would like to point you to a newsgroup discussion thread about this
subject:

-1 as a Prime (geometry-research, Math Forum)
http://mathforum.org/kb/message.jspa?messageID=1093170

A lot of this thread is useless, but the first (and later) messages by
John Conway do explain why he considers -1 a prime, and why this is
useful in his opinion.

I hope you appreciate this addition.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/ "

I no longer have this record of Dr. Conway's justification and was wondering if anyone does..in some way, shape or form.

Does anyone have an example of an "arithmetic contradiction" if -1 is considered prime? Do any mathematics get broken (or fixed)?
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Old 2020-10-29, 07:56   #2
axn
 
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Quote:
Originally Posted by jwaltos View Post
Does anyone have an example of an "arithmetic contradiction" if -1 is considered prime? Do any mathematics get broken (or fixed)?
Unique factorization will be broken if -1 is considered a prime. Just like 1 is given a special status of "unit", -1 should be given a special status and leave it at that, rather than trying to shoehorn it into existing framework.
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Old 2020-10-29, 12:54   #3
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Euclid used number to mean an integer greater than 1, while 1 was the unit. Thus, a prime number was by definition greater than 1.

If you want to bring negative numbers into the picture, the obvious generalization is that of an ideal. However, in any ring R, the ideal (-1) generated by -1 is the whole ring R.

The only context involving "primes" that comes to mind in which sign comes into play, is with a usage of "prime" related to what are called "valuations." If K is a number field, an embedding \sigma :\;K\;\rightarrow\;\mathbb{C} into the complex numbers is an "infinite prime," and an embedding \sigma :\;K\;\rightarrow\;\mathbb{R} is a "real infinite prime."

There is a concept of "multiplicative congruences" which is used in class field theory. They are expressed

x\;\equiv\;1\;\text{mod}^{\times}\;M

where M is a "modulus" consisting of a integral ideal (an ideal in the ring of algebraic integers R of K) and, possibly, a set of "real infinite primes" of K.

It includes the convention that, if p_{\infty} is a "real infinite prime" of K, the notation

x\;\equiv\;1\;\text{mod}^{\times}\;p_{\infty}

means that in the embedding \sigma :\;K\;\rightarrow\;\mathbb{R} defined by p_{\infty}, \sigma(x)\;>\;0.

With the rational integers \mathb{Z}, this allows a distinction between prime ideals having (say) a generator congruent to 1 (mod 4), and those with a positive generator congruent to 1 (mod 4). If p is any odd prime, (p) has a generator congruent to 1 (mod 4) since either p or -p is congruent to 1 (mod 4). But (p) only has a positive generator congruent to 1 (mod 4) when p is a prime number congruent to 1 (mod 4).

That's not anything like "-1 is a prime number," but it's as close as I can get with the math I am familiar with.
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Old 2020-10-29, 17:52   #4
kriesel
 
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Don't have to know much math to know -1 is more special than the garden variety prime, as are 1 and 0.
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Old 2020-10-29, 22:26   #5
jwaltos
 
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Quote:
Originally Posted by axn View Post
Unique factorization will be broken if -1 is considered a prime. Just like 1 is given a special status of "unit", -1 should be given a special status and leave it at that, rather than trying to shoehorn it into existing framework.
I appreciate your response and I'm glad that you phrased your reply this way. Two things, nothing should be "given" mathematically..given by whom, what, why...and second, "shoehorning" ["embedding" is more appropriate but your word has a stronger psychological impact] is like data manipulation akin to a lawyer presenting a case to the best of their ability by "truthful" misrepresentation and skewed semantic manipulation.
For pragmatic purposes, I try to understand how certain people think and have thought since all of mathematics (in the most general sense) was and is created by human thought. As an analogy to [one of] my assessments of questions like the subject of this thread, I specialized in mountain combat years ago (jungle and desert as well) and the following link describes a problem and an approach:
https://www.livius.org/sources/conte...-sogdian-rock/ (the episode on Ancient Assassins is more graphic).
You need to have " lots of skin in the game" , awareness of the situation and the imagination to cope and excel..in some ways like poker as well..belief systems play a major part. I hope my oblique response makes it clear that I dismiss your premise and conclusion and that your assessment of my approach wasn't ever a consideration.

Axn, you're obviously aware of the Riemann Hypothesis as am I (I was corrected within this forum regarding the RH by an astute member who caught my mistake in a prior post which is why I like this forum.) so I have a question for you that relates to the thread subject, "What is your interpretation of the "1/2 + i" within Riemann's expression?

Sardonicus, thanks for your analysis. As always, it's very well thought out.

Cheers.

As an afterthought, here is a link to a YouTube video and a proper preparation to this type is thinking is reviewing all of Ramanujan's notebooks and possibly some of his early published papers on Bernoulli numbers... Many of the papers I had collected over the years have been lost but memory and a rebuilt virtual library do help.
https://www.youtube.com/watch?v=jcKRGpMiVTw

Last fiddled with by jwaltos on 2020-10-30 at 00:24 Reason: Afterthought
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Old 2020-10-30, 02:22   #6
CRGreathouse
 
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In number theory it's all a matter of definition, of course. But in abstract algebra this is life or death -- you have lots of groups, rings, fields, etc. with units and it's important to agree on what precisely they are, and it's certainly not the case that (1) all objects have units or that (2) all units are 1 [in rings and fields, where such notion makes sense].

By any reasonable algebraic definition, 1 and -1 are units and neither is prime or irreducible. (Yes, in the integers, every prime is irreducible, but that's not true in general.)
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