Joerg Sonnenberger <joerg%britannica.bec.de@localhost> writes: > 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. Who told that? Please, avoid referring to "I was taught this way and refuse to learn anything else." What is the basis under definition you use? >> >> 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. Again, avoid referring to "I was taught this way and refuse to learn anything else." In most frequent case there's no inverse function, this doesn't stop you From calculating inverse trigonometric and hyperbolic functions using arbitrary branch cuts. Both, N and Z allow consistent division by zero. Yes, you weren't told that in your elementary school. That many books on arithmetic and algebra don't consider this may have quite another reasons like those mentioned by Quentin. >> > 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. You have stripped part of question to make it easier to attack. I didn't hear any rationale for your rejection of 1 and 0 as prime numbers. I count it as you don't have any. Now, please, revert your changes and leave implementation of correct factorization to others. -- HE CE3OH...
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