Joerg Sonnenberger <joerg%britannica.bec.de@localhost> writes:
> On Mon, May 17, 2010 at 12:47:53AM +0400, Aleksej Saushev wrote:
>> Joerg Sonnenberger <joerg%britannica.bec.de@localhost> writes:
>>
>> > On Sun, May 16, 2010 at 08:12:57PM +0400, Aleksej Saushev wrote:
>> >> I have spent several years studying math and I say that this approach is
>> >> wrong
>> >> both in mathematical and procedural sense. It is perfectly valid to count
>> >> 0,
>> >> +1 and -1 as prime numbers and thus factor any finite ones, it just
>> >> involves
>> >> a bit more math than it is taught in school.
>> >
>> > Where? This is not about what is taught in school, but the generally
>> > accepted definition of "prime".
>>
>> It is generally accepted that everything happens on the will of Allah,
>> but we're not on theological dispute. What matters here is that you
>> violate established procedures on disputed issues in case where it is
>> more or less clear way how to resolve them. I find it bad attitude of yours
>> that you:
>> 1. "Fix" disputed issues before the dispute is resolved thus
>> forcing others to follow "look, it's already there" approach.
>> 2. "Fix" them contrary to what would be consensus.
>> 3. Use arguments ex cathedra to support your actions without critically
>> considering them.
>> 4. Habitualy manage things this way.
>
> You still have done nothing to support your argument. I am still waiting
> to see any sort of authoritive reference why -1, 0 or 1 are primes or
> why this numbers have a non-empty, finite, unique set of prime factors.
> Ad hominem is not changing the lack of support.
Consider natural numbers as finite cardinals, the way set theorists do.
Factorization of a number a is non-empty set {<f_i, n_i>} (f_i are distinct)
such that product f_i^n_i = a. It is easy to see that there exist numbers
which occur in any their factorization. These are called "prime" numbers.
And zero and unit are among them. Note that you have to introduce
"non-empty" to treat unit rather than zero. That makes zero no less
prime than anything else and even a bit more prime than unit.
When you construct integer numbers, you see that the same applies to "-1".
That you prefer to choose "2" as prime rather than "-2" happens due to the
wish to preserve (where applicable) theorems when reducing back to natural
numbers.
Prime factorization is factorization of minimal cardinality where all
factors are prime (and orders are minimal possible, if you use model as
above, or sum of orders, if you like that better).
Note that you can model factorization any way you like it, e.g. using
mapping of starting interval, if you like that approach more.
Though you may have to define equality on factorizations and consider
equality classes.
Using this kind of constructions is a homework for math students
when they study set theory. Yes, this isn't taught in elementary school,
perhaps this may be told only in special courses devoted to foundations
of mathematics or to mathematical logic. I don't know the structure of
math education there.
Anyway this construction covers all integers and is superset of
what you were taught in school.
--
HE CE3OH...
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