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Title: Math Forum - Ask Dr. Math

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Why is 1 Not Considered Prime?

Date: 20 Mar 1995 12:22:37 -0500
From: ioostind at cln.etc.bc.ca (Ian Oostindie)
Subject: Why 1 is prime

        My friend, Roger Gillies told me he received some 
useful math information from you and gave me your e-mail 
address.  I thought of you when a grade six student stumped 
me with a classic.  Well, at least a classic in my mind.

        Just recently a grade six student asked me "Why is 1 
not considered prime?"  I tried to answer but as usual 
could not since I do not understand this either.  I thought 
it may lie in the fact that "we" don't use the true definition 
or we are interpreting it wrong.  A prime is normally 
described as a number that can be expressed by only one and 
itself.  We exclude all non-natural numbers from the set that 
we will be working on and then everything is fine except for 
when we work with 1.

        1 = 1 x 1.  That is, one equals 1 times itself and there 
is no other combination.

        Now to the grade six student in Faro Yukon, I said 
there may be a small print clause in the contract with the 
math gods that says you can only write it once since 1 also 
equals 1x1x1x1x...   This would not work for other primes 
such as two: 2 does not equal 1x2x2x2x...  Likewise, 3 does 
not equal 1x3x3x3x...

        Patterns are very important to mathematics, I further 
explained, and this is a pattern I see being broken.  I showed 
this in a slightly different way to the grade sixer but in 
essence the same.

        My question to you, Dr. Math, is what is the small print 
in the contract with the Math gods and how do we explain it 
to the grade six kids that are supposed to know it?

        Thank you very much for any consideration you make.

Date: 25 Mar 1995 16:21:45 -0500
From: Dr. Ken
Subject: Re: Why 1 is prime

Hello there!

Yes, you're definitely on the right track.  In fact, it's precisely 
because of "patterns that mathematicians don't like to break" 
that 1 is not defined as a prime.  Perhaps you have seen the 
theorem (even if you haven't, I'm sure you know it intuitively) 
that any positive integer has a unique factorization into primes.  
For instance, 4896 = 2^5 * 3^2 * 17, and this is the only possible 
way to factor 4896.  But what if we allow 1 in our list of prime 
factors?  Well, then we'd also get 1 * 2^5 * 3^2 * 17, and 
1^75 * 2^5 * 3^2 * 17, and so on.  So really, the flavor of the 
theorem is true only if you don't allow 1 in there.

So why didn't we just say something like "a prime factorization
is a factorization in which there are no factors of 1" or 
something?  Well, it turns out that if you look at some more 
number theory and you accept 1 as a prime number, you'd have 
all kinds of theorems that say things like "This is true for all 
prime numbers except 1" and stuff like that.  So rather than 
always having to exclude 1 every time we use prime numbers, 
we just say that 1 isn't prime, end of story.

Incidentally, if you want to call 1 something, here's what it is: 
it's called a "unit" in the integers (as is -1).  What that means is 
that if we completely restrict ourselves to the integers, we use 
the word "unit" for the numbers that have reciprocals (numbers 
that you can multiply by to get 1).  For instance, 2 isn't a unit, 
because you can't multiply it by anything else (remember, 1/2 
isn't in our universe right now) and get 1.  This is how we 
think about things in Abstract Algebra, something sixth graders 
won't need to worry about for a long time, but I thought I'd 
mention it.

-Ken "Dr." Math

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