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### Is -1 Prime?

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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/
```
Associated Topics:
High School Number Theory

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