Is -1 Prime?

Date: 10/20/2000 at 09:57:04
From: Tom Nesbitt
Subject: Is -1 Prime?

My class discussed prime numbers. The class noted that -1 has exactly 
two factors (namely 1 and -1) and asked why it wasn't listed as a 
prime number. I reiterated the definition, but they weren't satisfied. 
Do you have any further information on this topic?

Date: 10/20/2000 at 12:28:25
From: Doctor Peterson
Subject: Re: Is -1 Prime?

Hi, Tom.

Did you check our FAQ on Primes, or search our archives for "negative 
prime"? Here is one answer I found there:

   Why Aren't There Negative Prime Numbers?
   http://mathforum.org/dr.math/problems/lopez.12.10.99.html   

I assume that your definition of prime includes the word "positive," 
as in our FAQ. If the students are asking why the definition doesn't 
allow for negative primes, then this and other answers we have given 
should help.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 10/20/2000 at 13:07:15
From: Tom Nesbitt
Subject: Re: Is -1 Prime?

Thanks for writing back, Dr. Peterson.

After writing you, students found the FAQ you enclosed and were 
appeased. The book we are using, Glencoe, defines a prime number as 
"A number that has exactly two factors, 1 and the number itself." 
I showed them a couple of other definitions that include natural 
numbers, counting numbers, or positive integers. They now want to 
write to the company and suggest a correction. One of the authors 
teaches in nearby Unionville-Chaddsford, so we may make this a 
letter-writing lesson too.

Thanks for your help.

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/