1 and 0

Date: Fri, 4 Nov 94 08:33:09 +1000
From: - -
Subject: maths problems

Is 1 or 0 a prime number ?

Why ?

Please reply ASAP to form 2 maths group1 at 
St. Michael's  Grammar School 
Melbourne Australia

Date: Thu, 3 Nov 1994 19:42:56 -0500 (EST)
From: Dr. Ken (dr.math@mathforum.org)
Subject: Re: maths problems
To: stmg@ozemail.com.au (- -)

Hello Melbourne!

Technically, neither 1 nor zero is a prime number.  It is easiest to see 
why zero isn't:  since a prime number is only divisible by one and itself, 
let's find all the divisors of zero.

Well, since 0 x 1 = 0, and 0 x 2 = 0, and 0 x 3 = 0, and so on, all these
numbers divide zero, i.e. zero is divisible by every positive integer.  So
it isn't a prime number.

As for 1, you might want to call it a prime number, since it really _is_
divisible by only one and itself.  But then you run into some problems.  
For instance, you may know that every positive integer can be factored 
into the product of prime numbers, and that there's only one way to do 
it  for every number.  For instance, 280 = 2x2x2x5x7, and there's only 
one  way to factor 280 into prime numbers.  But if you let 1 be a prime, 
then  you can get the following factorizations: 1x1x1x2x2x2x5x7, 
1x2x2x2x5x7,  and so on.  The factorization is no longer unique.  

Furthermore, there are a whole bunch of theorems in Number Theory 
that tell you something about prime numbers.  But most of these 
theorems just flat out ain't true for the number 1.  So in light of these 
facts, we just declare the number 1 to not be a prime.

So that's why we don't WANT 1 to be a prime.  Mathematicians have summarized
this in a nice neat definition: a prime number is a positive integer which
has exactly 2 different positive integers that divide it evenly - no more
and no fewer.


-Ken