tag:blogger.com,1999:blog-12535639.post111835377940689791..comments2020-07-12T09:58:25.710-07:00Comments on Fermat's Last Theorem: Unique FactorizationLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-12535639.post-32563852861921934342015-01-25T11:59:00.011-08:002015-01-25T11:59:00.011-08:00Hi T0hierry,
It is similar to orthonormal basis i...Hi T0hierry,<br /><br />It is similar to orthonormal basis in the sense that an integer can be represented as a set of primes.<br /><br />Unique factorization means that the integers can only be represented in one, unique way.<br /><br />Not all integers have unique factorization. For example, the cyclotomic integers do not which resulted in a mistaken proof of Fermat's Last Theorem by Gabriel Lame.<br /><br />Larry Freemanhttps://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-46179073459898263832015-01-25T02:02:52.813-08:002015-01-25T02:02:52.813-08:00Hi Larry,
I'm reading your very informative p...Hi Larry,<br /><br />I'm reading your very informative post as a physicist. Is uniqueness of factorization similar in concept to that of orthonormal basis?t0hierryhttps://www.blogger.com/profile/09222155493420841770noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-17079634397674840682008-01-03T03:49:00.000-08:002008-01-03T03:49:00.000-08:00Hi Tony,Thanks very much for your comment! You ar...Hi Tony,<BR/><BR/>Thanks very much for your comment! You are absolutely correct that unique factorization as a concept has been well known since <A HREF="http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX14.html" REL="nofollow">Euclid</A>. The sentence as stated is incorrect.<BR/><BR/>My intention in the blog is to call out the importance of establishing unique factorization for the algebraic integers. This is what Gauss was the first to do and that is why we call Z[i], Gaussian Integers.<BR/><BR/>In light of your comments, I have updated my blog to make my main point more clear.<BR/><BR/>Regards,<BR/><BR/>-LarryLarry Freemanhttps://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-66410127631882346352008-01-03T03:22:00.000-08:002008-01-03T03:22:00.000-08:00Hi, Larry. Your blog gives the impression that Gau...Hi, Larry. Your blog gives the impression that Gauss was the first person to realise the need to prove unique factorisation for integers. You're only two thousand years out! Euclid saw the need for a proof and gave it as Proposition 14 of Book IX of his Elements.<BR/><BR/>Tony SudberyTony Sudberyhttps://www.blogger.com/profile/14703962172816776765noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-1158864871192069382006-09-21T11:54:00.000-07:002006-09-21T11:54:00.000-07:00Hi Jose,Thank you very much for your comments. I h...Hi Jose,<BR/><BR/>Thank you very much for your comments. <BR/><BR/>I have added a note to the blog and also add a reference to the MathWorld article which discusses your point.<BR/><BR/>In the future, I will write a blog on this topic.<BR/><BR/>Thank you very much for noticing this and bringing it up! :-)<BR/><BR/>-LarryLarry Freemanhttps://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-1158860266152420562006-09-21T10:37:00.000-07:002006-09-21T10:37:00.000-07:00Hi Larry:You use a quite unconventional definition...Hi Larry:<BR/><BR/>You use a quite unconventional definition for the norm. I know that is sometimes used (like in Hardy&Wright), mostly referred to gaussian integers, but it's a bad choice because it doesn't fulfill all the properties traditionally asigned to a norm, and the complex norm has a square root: ||a+bi|| = Sqrt(a^2+b^2).<BR/><BR/>Regards. Jose BroxJose Broxhttps://www.blogger.com/profile/11420726252242217410noreply@blogger.com