tag:blogger.com,1999:blog-264226589944705290.post2677501571609792133..comments2021-12-14T05:53:12.175-08:00Comments on God Plays Dice: A thing I didn't know about Pythagorean triplesMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-264226589944705290.post-70838539379997549542008-03-18T08:09:00.000-07:002008-03-18T08:09:00.000-07:00what about these?for (3,4,5)=> 3^2 = 9 = 4 + 5(5,1...what about these?<BR/><BR/>for (3,4,5)=> 3^2 = 9 = 4 + 5<BR/>(5,12,13)=> 5^2 = 25 = 12 + 13<BR/>(7,24,25)=> 7^2 = 49 = 24 + 25<BR/>...<BR/>(a,b,c)=> a^2 = b + c<BR/>a = odd numbers<BR/>b = (((a^2)-1)/2)<BR/>c = (((a^2)+1)/2)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-37175993909954599912008-02-19T01:11:00.000-08:002008-02-19T01:11:00.000-08:00You can also generalise to show that if you multip...You can also generalise to show that if you multiply numbers which are the sum of two squares you get a number which is the sum of two squares in two different ways.<BR/><BR/> a = x.x + y.y<BR/> b = s.s + t.t<BR/><BR/> z = x + iy => a = zz*<BR/> u = s + it => b = uu*<BR/><BR/> c = ab = z(z*)u(u*) = (zu)(zu)* = p.p + q.q<BR/><BR/> with p = Re(zu) = xs - yt<BR/> q = Im(zu) = xt + sy<BR/><BR/> but also<BR/><BR/> c = ab = z(z*)u(u*) = (zu*)(zu*)* = p'.p' + q'.q'<BR/><BR/> with p' = Re(zu*) = xs - yt<BR/> q' = Im(zu*) = xt + sy<BR/><BR/>and you can do similar tricks with quarternions and octonions. A lot of number theory is based on further generalisations of these formulae to other ringsPeteHHhttps://www.blogger.com/profile/08545699636937390967noreply@blogger.com