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Re: prime numbers and African artifact
David Ullrich (ullrich@math.okstate.edu)
15 Jul 1995 17:44:23 GMT
Daniel Kian Mc Kiernan <dmckiern@weber.ucsd.edu> wrote:
>On 13 Jul 1995, David Ullrich wrote:
>
>] Daniel Kian Mc Kiernan <dmckiern@weber.ucsd.edu> wrote:
>]
>]> I'm familiar with three definitions of "prime number".
>]>
>]> [1] A positive integer divisible only by itself and by 1.
>]>
>]> [2] Same as [1] except that the number must also be greater than 1.
>]>
>]> [3] Same as [1] or [2] except that the number must also be greater
>]> than 2.
>]>
>]> For my part, I don't care for definitions [2] or [3].
>]
>] Hmm. How do you feel about the uniqueness of prime factorization
>] for positive integers? A lot of us are for it.
>
>Sorry, you can't quite =have= that, as 1 is a positive integer.
>
>When you sit down and state what you =can= have, you should see that
>the matter comes down to =where=, rather than =whether=, you use an
>expression equivalent to "except 1".
>
Sorry to answer this twice, but in case you felt yesterday's "Oh
yeah, 1 is the product of no primes at all" was silly and/or still
required a special definition:
It's not hard to give a statement of the unique prime
factorization that applies to every positive integer (including 1),
without needing to worry about what the product of no primes at all
is. Let p_1,p_2,... denote the primes. THM If n is any positive
integer then there exists a unique sequence of non-negative integers
e_1,... (all but finitely many of which are 0) such that
n=product(p_j^e_j:j=1..infinity).
Sure enough even 1 has a unique prime factorization - that
was close.
Dave Ullrich
(What does it matter anyway? If we say 1 is prime then 2 no
longer has a unique prime factorization - most people regard this as
a sufficient reason to declare 1 a non-prime.)
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