tag:blogger.com,1999:blog-12535639.post4185154945973185087..comments2024-02-26T06:55:41.876-08:00Comments on Fermat's Last Theorem: Starting a new blog on algorithmsLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-12535639.post-44608876572223183952017-10-03T01:45:23.513-07:002017-10-03T01:45:23.513-07:001. There is another explanation of a simple proof ...1. There is another explanation of a simple proof of Fermat’s last theorem as follows:<br /><br />X^p + Y^p ?= Z^p (X,Y,Z are integers, p: any prime >2) (1)<br /><br />2. Let‘s divide (1) by (Z-X)^p, we shall get:<br /><br />(X/(Z-X))^p +(Y/(Z-X))^p ?= (Z/(Z-X))^p (2)<br /><br />3. That means we shall have:<br /><br />X’^p + Y’^p ?= Z’^p and Z’ = X’+1 , with X’ =(X/(Z-X)), Y’ =(Y/(Z-X)), Z’ =(Z/(Z-X)) (3)<br /><br />4. From (3), we shall have these equivalent forms (4) and (5):<br /><br />Y’^p ?= pX’^(p-1) + …+pX’ +1 (4)<br />Y’^p ?= p(-Z’)^(p-1) + …+p(-Z’) +1 (5)<br /><br />5. Similarly, let’s divide (1) by (Z-Y)^p, we shall get:<br /><br />(X/(Z-Y))^p +(Y/(Z-Y))^p ?= (Z/(Z-Y))^p (6)<br /><br />That means we shall have these equivalent forms (7), (8) and (9):<br /><br />X”^p + Y”^p ?= Z”^p and Z” = Y”+1 , with X” =(X/(Z-Y)), Y” =(Y/(Z-Y)), Z” =(Z/(Z-Y)) (7)<br /><br />From (7), we shall have:<br /><br />X”^p ?= pY”^(p-1) + …+pY” +1 (8)<br />X”^p ?= p(-Z”)^(p-1) + …+p(-Z”) +1 (9)<br /><br />Since p is a prime that is greater than 2, p is an odd number. Then, in (4), for any X’ we should have only one Y’ (that corresponds with X’) as a solution of (1), (3), (4), (5), if X’ could generate any solution of Fermat’s last theorem in (4).<br /><br />By the equivalence between X’^p + Y’^p ?= Z’^p (3) and X”^p + Y”^p ?= Z”^p (7), we can deduce a result, that for any X” in (8), we should have only one Y” (that corresponds with X’’ ) as a solution of (1),(7),(8),(9), if X” could generate any solution of Fermat’s last theorem.<br /><br />X” cannot generate any solution of Fermat’s last theorem, because we have illogical mathematical deductions, for examples, as follows:<br /><br />i)In (8), (9), if an X”1 could generate any solution of Fermat’s last theorem, there had to be at least two values Y”1 and Y”2 or at most (p-1) values Y”1, Y”2,…, Y”(p-1),<br />that were solutions generated by X”, of Fermat’s last theorem. (Please note the even number (p-1) of pY”^(p-1) in (8)). But we already have a condition stated above, that for any X” we should have only one Y” (that corresponds with X”) as a solution of (1),(7),(8),(9), if X” could generate any solution of Fermat’s last theorem.<br />Fermat’s last theorem is simply proved!<br /><br />ii)With X”^p + Y”^p ?= Z”^p, if an X”1 could generate any solution of Fermat’s last theorem, there had to be correspondingly one Y” and one Z” that were solutions generated by X”, of Fermat’s last theorem. But let’s look at (8) and (9), we must have Y” = -Z”. This is impossible by further logical reasoning such as, for example:<br /><br />We should have : X”^p + Y”^p ?= Z”^p , then X”^p ?= 2Z”^p or (X”/Z”)^p ?= 2. The equal sign, in (X”/Z”)^p ?= 2, is impossible.<br />Fermat’s last theorem is simply again proved, with the connection to the concept of (X”/Z”)^p ?= 2. Is it interesting?Anonymoushttps://www.blogger.com/profile/11885516103420681253noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-70841206317559640222008-05-26T13:18:00.000-07:002008-05-26T13:18:00.000-07:00The material you present is very interesting and u...The material you present is very interesting and useful to an amateur like me. Although I don't have much proficiency with algebraic number theory, I try to make up for it with empirically derived results pertaining to FLT. For example, Wieferich's criterion can be derived using the "pth power with respect to" concept (see my website at "http://home.graysoncable.com/dkcox"). I have a large amount of software that I'm willing to share with anyone who's interested.dkchttps://www.blogger.com/profile/08766673018639813247noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-32068424939936622008-03-22T12:08:00.000-07:002008-03-22T12:08:00.000-07:00Also, if you are interested in the history of Ferm...Also, if you are interested in the history of Fermat's Last Theorem.<BR/><BR/>In a Feb 2008 article, historians discovered additional papers by Sophie Germain on her efforts to solve Fermat's Last Therem.<BR/><BR/>Here's an <A HREF="http://www.sciencenews.org/articles/20080223/mathtrek.asp" REL="nofollow">article</A> on this development.Larry Freemanhttps://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-8940943144958959432008-03-22T12:05:00.000-07:002008-03-22T12:05:00.000-07:00In 2005, there was a possible generalization of An...In 2005, there was a possible generalization of Andrew Wile's result by<BR/><A HREF="http://plus.maths.org/latestnews/jan-apr05/serre/" REL="nofollow">Chandrashekhar Khare</A>. There's a link to his paper from the web site.<BR/><BR/>Here's <A HREF="http://www.hinduonnet.com/2005/04/25/stories/2005042506530100.htm" REL="nofollow">another article</A> on the same person.<BR/><BR/>I hope to cover this advance after I complete my analysis of Andrew Wile's second proof.<BR/><BR/>-LarryLarry Freemanhttps://www.blogger.com/profile/06906614246430481533noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-15623976242234356282008-03-21T10:42:00.000-07:002008-03-21T10:42:00.000-07:00This may be off topic.It has been a while since re...This may be off topic.<BR/><BR/>It has been a while since reading anything about Fermat's last theorem. <BR/><BR/>If someone had an alternate solution compared to Wiles, any idea where he could go to submit the information?YellowJellyhttps://www.blogger.com/profile/11163225472723305848noreply@blogger.com