Factors and Primes

Date: 09/10/97 at 12:13:07
From: Jacob Dick
Subject: Factors and primes

I'm student teaching right now in a middle school and I know that a 
prime number is a number greater than 1, and has only 2 divisors, 1 
and itself. I was just curious... if they didn't make the stipulation 
greater than one, how much would that affect it? Wouldn't the only 
added number be negative one, since its factors would be 1 and -1?  

Date: 09/10/97 at 12:26:48
From: Doctor Rob
Subject: Re: Factors and primes

The Fundamental Theorem of Arithmetic, that every positive integer
can be uniquely factored into a product of powers of prime numbers,
would fail, since 3 = 1*3 = 1^2*3 = 1^3*3 = ... .  It would have to be
restated to say that every positive integer can be uniquely factored
into a product of powers of prime numbers different from 1.

Many other theorems would have to be restated to say things about
"prime numbers different than 1."  Example: every prime number
(different from 1) has exactly two positive divisors. It is just
simpler to exclude 1 from the set of prime numbers.

Furthermore, 1 is a member of a different class: the units. 
They are integers whose reciprocal is also an integer. They consist 
of 1 and -1 only. The set of integers (or, more generally, any ring) 
is generally split into four sets: the zero-divisors, the units, the
primes, and the composites. These are mutually exclusive and
collectively exhaustive.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/