Re: prime numbers and African artifact
Michael Jennings (M.J.Jennings@amtp.cam.ac.uk)
13 Jul 1995 20:06:05 GMT
In article <Pine.HPP.3.91.950713005007.12887Hfirstname.lastname@example.org>,
Daniel Kian Mc Kiernan <email@example.com> wrote:
>On Tue, 11 Jul 1995, Alistair J. R. Young wrote:
>>Rick Hawkins writes:
>>> But only half- credit, since it's the wrong answer. 1 is not prime.
>> Correct me if I'm wrong but if a prime number is only divisible by itself
>> and 1, 1 is prime. What else is it divisible by?
>I'm familiar with three definitions of "prime number".
> A positive integer divisible only by itself and by 1.
> Same as  except that the number must also be greater than 1.
> Same as  or  except that the number must also be greater
> than 2.
>For my part, I don't care for definitions  or .
Why not? Definition two is the only sensible one, as
using this definition it is possible to express any positive integer
other than one as a unique product of prime numbers. This is why prime
numbers are useful - this result isn't called the fundamental theorem of
arithmetic for nothing. In fact, in one way it is best to use as a
"The prime numbers are the set of positive integers such
that any positive integer other than one can be created as a unique
product of elements (which can be used more than once each) of the set"
This definition explains why we have such a thing as a 'prime
number'. Unfortunately this is a totally non-constructive definition.
Other equivalent definitions, such as (2) above are more useful in
determining such things as whether a number is prime, and what the
prime numbers actually are, and are therefore more common.
The number 1 is the multiplicative identity, something quite special
and very important, but something entirely different from a prime number.
Department of Applied Mathematics and Theoretical Physics
The University of Cambridge. firstname.lastname@example.org
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