Jim Choate wrote:
Forwarded message:
Date: Thu, 19 Nov 1998 11:50:34 -0600 Subject: Math Forum - Ask Dr. Math
<META NAME="title" CONTENT="Why is 1 Not Considered Prime?">
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...
Whether the 1 is there or not is irrelevant,
3x3x3x3... is not 3 in the first place.
3x1x1x1x1.... IS 3.
Patterns are very important to mathematics, I further explained, and this is a pattern I see being broken.
Date: 25 Mar 1995 16:21:45 -0500 From: Dr. Ken Subject: Re: Why 1 is prime
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.
This definition of a prime has one serious drawback.
It ignores the fundamental identity theorem of arithmatic:
1 * n = n
it is not a theorem, it is a part of the definition of multiplication. It is, in truth, arithmetics. igor
So, as a result we're in the interesting and potentialy untenable situation of defining a identity theorem, base our math on it, and then come along one day and say it doesn't apply anymore WITHOUT making any other changes to the structure of the theories....
This is VERY BAD science/math.
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