tag:blogger.com,1999:blog-2433841880619171855.post3237421481581731386..comments2024-03-27T21:09:44.320+00:00Comments on Pat'sBlog: An Interesting Observation, and a ProblemUnknownnoreply@blogger.comBlogger5125tag:blogger.com,1999:blog-2433841880619171855.post-2703546370657964322009-09-26T07:19:03.791+01:002009-09-26T07:19:03.791+01:00And by the way, you're wrong. You do say 0^0 ...And by the way, you're wrong. You do say 0^0 = 1, not undefined. Unless when you say "The binomial theorem states ..." you include "unless x+y = 0 and n = 0, in which case ..."<br /><br />Same with people who claim they define trapezoid as a quadrilateral having exactly one pair of parallel sides. They all still say the "trapezoid rule" for integrals, not the "trapezoid or rectangle", and they find the area of those trapezoids, not distinguishing two cases depending on whether f(x0) and f(x1) are the same ...<br /><br />There are other places where theorems would have annoying exceptions if 0^0 were anything but 1. I have yet to find any meaningful theorem where I'd be wrong to define it as 1.<br /><br />I was even thinking that something trivial like 0^x = 0 might be such a theorem, but it's not. I say "0^x = 0 for all x > 0" and the people who think 0^0 is undefined say exactly the same thing.Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-20022526719374186042009-09-26T07:15:15.909+01:002009-09-26T07:15:15.909+01:00No, there aren't any more. At the end of the ...No, there aren't any more. At the end of the OEIS entry you can see the keyword "full" which means that all the terms are listed here (and "fini" which means there are finitely many terms).Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-63169719547596860622009-09-25T17:25:22.537+01:002009-09-25T17:25:22.537+01:00Well I leave out 0^0 as a case (I just call it und...Well I leave out 0^0 as a case (I just call it undefined) but I think it is remarkable that there is actually another, and that it is SOOOOOO very large... 438,579,088l... so there probably are more...but gosh, how big must they be... apparently even the on-line inter sequence didn't find the next one..Pat's Bloghttps://www.blogger.com/profile/15234744401613958081noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-30615913956578255652009-09-25T08:36:20.512+01:002009-09-25T08:36:20.512+01:00By the way, http://www.research.att.com/~njas/sequ...By the way, http://www.research.att.com/~njas/sequences/A046253 will spoil your fun if you are interested in working further on this problem.Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-85434708621357215502009-09-25T08:35:07.114+01:002009-09-25T08:35:07.114+01:00Other than 1, and possibly 0 depending on what you...Other than 1, and possibly 0 depending on what you think 0^0 is equal to, there aren't many more. There's a not-too-hard proof that there can only be finitely many. A computer search up into the multi-millions doesn't find any more besides 3435.<br /><br />(I assumed 0^0 = 1 for all my work)Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.com