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Re: CVS commit: src/games/factor

On Tue, May 18, 2010 at 12:35:16AM +0400, Aleksej Saushev wrote:
> >> It is easy to see that there exist numbers
> >> which occur in any their factorization. These are called "prime" numbers.
> >
> > That's not the definition of prime elements algebra uses, which is the
> > relevant field here. The algebraic definition of prime explicitly
> > excludes units.
> There's no algebra here, it is arithmetics, algebra may be relevant here,
> but it isn't to a large extent.

The arithmetic definition of prime doesn't agree with you either.

> >> 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.
> >
> > The definition of prime also excludes 0 for the simple reason that 0
> > doesn't have any divisor in an integral domain like we are using here.
> Who are "we"? We're using "product decomposition," and 0 can be decomposed.
> Also 0 can be divisor and is divisor of 0, that you restrict yourself to
> elementary school definition of divisor proves my argument that you don't
> evaluate ex cathedra arguments you were once told.

The domain that is used by factor is either N or Z. Both don't have any
0 divisors. I don't know what approach to math you are using, but it
obviously still doesn't agree with the commonly accepted definitions of
either. Again, where is the consistent set of definitions you are using?
I dare to say that what you are saying violates all modern books on
arithmetic and algebra, but I am waiting to be proven wrong.

> > 2 and -2 are both primes and for that reason the factorization in Z is
> > only unique up to units.
> Oh, and in N it isn't?

N has only one unit, so yes, it is unique. But you obviously have a very
unique definition of prime, that has randomly other properties. As you
can't agree to use the definitions that are generally used in the field
of math, this discussion is as pointless as trying to argue with a
religious believer about God.


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