Re: Math Forum - Ask Dr. Math (fwd)

[ichudov@Algebra.COM (Igor Chudov @ home)]

Jim Choate wrote:
> 
> 
> Forwarded message:
> 
> > Date: Thu, 19 Nov 1998 11:50:34 -0600
> > Subject: Math Forum - Ask Dr. Math
> 
> > http://forum.swarthmore.edu/dr.math/problems/1isprime.html
> 
> > <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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	- Igor.