Re: prime numbers and African artifact

Daniel Kian Mc Kiernan (
Thu, 13 Jul 1995 21:25:38 -0700

On 13 Jul 1995, Michael Jennings wrote:

> Daniel Kian Mc Kiernan <> 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].
> 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.

No one is calling the importance of the theorem into question. It
is only insofar as how a definition of "prime" affects the felicity
of expression that the theorem can here be brought into play. NOw,
you didn't give a formal statement of the theorem, but nonetheless
look at what you did say, and how you said it. It really comes down
to nothing more than a question of where one sticks in "except one"
(or the equivalent) in the expression.

> In fact, in one way it is best to use as a
> definition:
> "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'.

But it really says =nothing= about a superiority of definition [2]
(or whatever) over any other definition.

> 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.

I take it that you're here simply restating your thesis, rather than
begging the question.

It's always Dark. Light only hides the Darkness.

Daniel Kian Mc Kiernan (619) 535 - 0546 <75013,676>