Re: Prime?

From: John Morrison (john.morrison_at_tesco.net)
Date: 12/25/04 

Date: Sat, 25 Dec 2004 01:11:32 -0000

Phil Carmody wrote:
> hrubin@odds.stat.purdue.edu (Herman Rubin) writes:
>
>> In article <AkYyd.556556$wV.195871@attbi_s54>,
>> robert j. kolker <nowhere@nowhere.com> wrote:
>>
>>
>>> Hydrokineticman wrote:
>>
>>>> Is 1 prime or composite?
>>>> What about 0?
>>>> What do you think?
>>
>>> By convention 1 is not a prime. That is a matter of definition.
>>
>> This is not definition; 1 is a unit.
>>
>> Multiplication or
>> division by a unit does not change primality.
>
> Lehmer called 1 a prime. If primality were to be defined to not
> exclude units, then obviously all units are prime, and thence
> multiplication or division by a unit would still not change
> primality. So your argument becomes void.
>
> Phil

    Poor old Lehmer! Ultimately, one has to have a natural number divisible
_uniquely_ by the (usual definition of) primes. If unity (one) be prime,
then (rather obviously) /every/ natural number is divisible by one - and not
only that: any number of times one wishes.

    The _only_ definition of prime which results in _unique_ factorization
is this:

Let N be a natural number. N is prime if _and only if_ it has two /distinct
factors/: unity and itself (/i.e./ N). If unity be allowed as a prime,
unique factorization breaks down.

    Sorry if this idea seems *too* obvious. Nevertheless, what I have just
suggested amounts to the only consistent definition of primes.

Given the time, I will _merely_ add: Merry Christmas and a Happy New Year!

John
johnDOTmorrisonATtescoDOTnet 

-- 
"Why do they sterilize the needle for lethal injections?" - Mark Ayliffe