tag:blogger.com,1999:blog-12604614.post111566336205858097..comments2017-03-10T02:08:06.554-08:00Comments on Math Refresher: Fundamental Theorem of ArithmeticLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comBlogger13125tag:blogger.com,1999:blog-12604614.post-35365566854464550522014-12-16T21:20:47.779-08:002014-12-16T21:20:47.779-08:00Hi Mr Freeman,
Theorem 1: Division Algorithm for ...Hi Mr Freeman,<br /><br />Theorem 1: Division Algorithm for Integers<br />(4e)<br /><br />But if b is negative, then a - b(c+1) = d - b is more than d, isn't it?<br /><br />-----------------<br />(5b)<br /><br />d - D should be D - d.<br /><br />----------------<br />(5d)<br /><br />Is it better to write 0 < D - d < b instead? Otherwise, D - d could be kb where k is a negative integer.<br /><br />And maybe also include d < D in (5a) to make 0 < D - d in (5d) more obvious?2j5khttp://www.blogger.com/profile/15557930348330564784noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-88997879355230587472014-12-16T21:19:12.341-08:002014-12-16T21:19:12.341-08:00This comment has been removed by the author.2j5khttp://www.blogger.com/profile/15557930348330564784noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-23864752319052344182014-12-16T21:16:23.254-08:002014-12-16T21:16:23.254-08:00This comment has been removed by the author.2j5khttp://www.blogger.com/profile/15557930348330564784noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-46548565044971691762011-08-22T06:11:54.942-07:002011-08-22T06:11:54.942-07:00Hi Gustolandia,
b > 0 doesn't need to be p...Hi Gustolandia,<br /><br />b > 0 doesn't need to be proved because c can be negative.<br /><br />In that case, a - bc is still >= 0 even when b is negative.<br /><br />-LarryLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-62041398803476928772011-08-22T03:06:58.136-07:002011-08-22T03:06:58.136-07:00Hi. b>0 is in need on the first proof.Hi. b>0 is in need on the first proof.Gustolandiahttp://www.blogger.com/profile/03209711837836686046noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-35346563186891125162010-06-01T16:07:08.225-07:002010-06-01T16:07:08.225-07:00Hi Scouse Rob,
Thanks for noticing. It's now...Hi Scouse Rob,<br /><br />Thanks for noticing. It's now been fixed.<br /><br />-LarryLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-61039280044552880342010-06-01T03:16:17.902-07:002010-06-01T03:16:17.902-07:00In Lemma 2 (5) a 'd' appears, which I thin...In Lemma 2 (5) a 'd' appears, which I think should be a 'b'.<br /><br />b(1) = b(as + pt) = b = abs + ptd<br /><br />RobScouse Robhttp://www.blogger.com/profile/00144454830208958210noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-51009507003440713732010-04-01T22:37:09.439-07:002010-04-01T22:37:09.439-07:00Hi Pat,
Thanks very much for noticing that! :-)
...Hi Pat,<br /><br />Thanks very much for noticing that! :-)<br /><br />I've fixed the typo.<br /><br />Cheers,<br /><br />-LarryLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-85882716560605972352010-04-01T16:52:59.082-07:002010-04-01T16:52:59.082-07:00I believe you have a typo under Theorem 1: Case II...I believe you have a typo under Theorem 1: Case II: (5)(a)<br /><br />It reads:<br />a = BC + D<br /><br />and I believe it's meant to read:<br />a = bC + D<br /><br />Not a huge deal, but it gave me a bit of a pause when I was reading over it.<br /><br />I'm enjoying your Fermat Blog and look forward to perusing this one more as well.<br /><br />Thanks for your efforts,<br />-PatP. B.http://www.blogger.com/profile/07168096867994037986noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-11463237423771361282008-02-28T17:03:00.000-08:002008-02-28T17:03:00.000-08:00Great question. The proof does not relate to 1. ...Great question. The proof does not relate to 1. 1 is the only positive integer that is not composed of primes. <BR/><BR/>In fact, we routinely say gcd(2,3)=1 to show that there is no common prime between two numbers.<BR/><BR/>So, to be more accurate, all integers greater than 1 are composed of a unique set of primes.<BR/><BR/>-LarryLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-20077557339627935712008-02-28T14:33:00.000-08:002008-02-28T14:33:00.000-08:00I really do hate to be the devils advocate here, b...I really do hate to be the devils advocate here, but your statement of the theorem indicates that EVERY integer is a product of a unique set of primes. Yet your proof only shows that numbers greater than 1 are products of a unique set of primes. How do you deal with the integer 1?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-12604614.post-1128906951859235992005-10-09T18:15:00.000-07:002005-10-09T18:15:00.000-07:00Hi Susan,Here's a link that shows many of his othe...Hi Susan,<BR/><BR/>Here's a link that shows many of his other works:<BR/>http://en.wikipedia.org/wiki/Euclid<BR/><BR/>-LarryLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12604614.post-1128905928883678142005-10-09T17:58:00.000-07:002005-10-09T17:58:00.000-07:00I am writing a paper on Euclid's other than his ge...I am writing a paper on Euclid's other than his geometry contributions. From the research I have done so far I have found the following contributions:<BR/>Number theory books 7,8,9 and proportional theory book 5, Elements of Music. Do you have other examples of contributions I can research. You seem to be a vast of knowledge!Susanhttp://www.blogger.com/profile/09396820736094644876noreply@blogger.com