tag:blogger.com,1999:blog-12535639.post2015903974289374371..comments2024-02-26T06:55:41.876-08:00Comments on Fermat's Last Theorem: Gauss: Seventeenth Root of Unity Expressed As RadicalsLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-12535639.post-2835524105460028322016-01-18T07:58:03.815-08:002016-01-18T07:58:03.815-08:00I will use "w" for the primitive 17th ro...I will use "w" for the primitive 17th root of unity (I don't know how to use Greek letters on this blog).<br /><br />The primitive 17th root of unity is cos(2*pi/17) + i sin(2*pi/17) [See an article on roots of unity for an explanation]. By DeMoivre's Theorem, w^n = cos(2n*pi/17)+i sin(2n*pi/17) So <br />w^16 = cos(32*pi/17) + i sin(32*pi/17) = cos(-2*pi/17) + i sin(-2*pi/17) since 32*pi/17 = -2*pi/17 [32*pi/17 measures the angle in the counter-clockwise direction, -2*pi/17 measures it in the clockwise direction] But recall that cos(-A) = cos(A) and sin(-A) = - sin(A), so<br />w + w^16 = 2*cos(2*pi/17) Bob Dillonhttps://www.blogger.com/profile/02286614703867794390noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-51085254996805892932016-01-17T00:45:15.960-08:002016-01-17T00:45:15.960-08:00Why is the real part of primitive seventeenth root...Why is the real part of primitive seventeenth root of unity half of Z-1? Is there any explanation or proof for this?Anonymoushttps://www.blogger.com/profile/02607302699430767519noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-43658175936209085692008-03-27T10:54:00.000-07:002008-03-27T10:54:00.000-07:00Actually, the final result is only the real part o...Actually, the final result is only the real part of zeta (which is all that's needed to construct the heptadecagon), and the final result cannot be found from step 20. The final result is half of z_1, and that is where the back substitution should start.Bob Dillonhttps://www.blogger.com/profile/01205833339772640410noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-86282356268814467942008-03-01T10:02:00.000-08:002008-03-01T10:02:00.000-08:00I have a question for Larry Freeman regarding this...I have a question for Larry Freeman regarding this proof. Larry, would you be willing to send your e-mail address to me at bdillon@aurora.edu ? My question is a bit long to post as a comment.<BR/><BR/>Thanks,<BR/><BR/>Bob Dillon<BR/>Aurora, IL<BR/>(I wasn't able to find any contact info on this blog site for you)Bob Dillonhttps://www.blogger.com/profile/01205833339772640410noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-16157304709917487632008-01-29T14:03:00.000-08:002008-01-29T14:03:00.000-08:00I just want to invite everyone to visit my homepag...I just want to invite everyone to visit my homepage.<BR/><BR/>http://www.geocities.com/jefferywinkler<BR/><BR/>Jeffery Winklerjefferywinklerhttps://www.blogger.com/profile/11937332813403141269noreply@blogger.com