For the past two years, I have been working exclusively on a math blog. It has really been great for me. It has been an opportunity to delve into the history of mathematics in the effort of reviewing great works of genius and try to put these achievements in the context of a very tough math problem: Fermat's Last Theorem.
In light of the great enjoyment I've had in running this blog, I've decided to start another blog. This one will be closer to my day job, software engineering.
The purpose of the new blog is to analyze computer algorithms in the same way that I have up to now been analyzing mathematical proofs. It will cover the classic works of algorithms and some contemporary algorithms.
For example, my first series of blogs will be on the $50,000 algorithm. This is the algorithm that has won the 2007 Progress Prize as part of the $1 million Netflix Prize contest. Details on this content can be found here.
I hope to keep up the same standards of rigorous logic as well as clarity and precision. If you have an interest in computer programming, I hope that you find my new blog as interesting complement to this math blog.
I plan to continue the Fermat's Last Theorem blog up until Andrew Wiles' second proof so if math is your thing, please continue coming here. :-)
Regards,
-Larry
Friday, February 22, 2008
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5 comments:
This may be off topic.
It has been a while since reading anything about Fermat's last theorem.
If someone had an alternate solution compared to Wiles, any idea where he could go to submit the information?
In 2005, there was a possible generalization of Andrew Wile's result by
Chandrashekhar Khare. There's a link to his paper from the web site.
Here's another article on the same person.
I hope to cover this advance after I complete my analysis of Andrew Wile's second proof.
-Larry
Also, if you are interested in the history of Fermat's Last Theorem.
In a Feb 2008 article, historians discovered additional papers by Sophie Germain on her efforts to solve Fermat's Last Therem.
Here's an article on this development.
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.
1. There is another explanation of a simple proof of Fermat’s last theorem as follows:
X^p + Y^p ?= Z^p (X,Y,Z are integers, p: any prime >2) (1)
2. Let‘s divide (1) by (Z-X)^p, we shall get:
(X/(Z-X))^p +(Y/(Z-X))^p ?= (Z/(Z-X))^p (2)
3. That means we shall have:
X’^p + Y’^p ?= Z’^p and Z’ = X’+1 , with X’ =(X/(Z-X)), Y’ =(Y/(Z-X)), Z’ =(Z/(Z-X)) (3)
4. From (3), we shall have these equivalent forms (4) and (5):
Y’^p ?= pX’^(p-1) + …+pX’ +1 (4)
Y’^p ?= p(-Z’)^(p-1) + …+p(-Z’) +1 (5)
5. Similarly, let’s divide (1) by (Z-Y)^p, we shall get:
(X/(Z-Y))^p +(Y/(Z-Y))^p ?= (Z/(Z-Y))^p (6)
That means we shall have these equivalent forms (7), (8) and (9):
X”^p + Y”^p ?= Z”^p and Z” = Y”+1 , with X” =(X/(Z-Y)), Y” =(Y/(Z-Y)), Z” =(Z/(Z-Y)) (7)
From (7), we shall have:
X”^p ?= pY”^(p-1) + …+pY” +1 (8)
X”^p ?= p(-Z”)^(p-1) + …+p(-Z”) +1 (9)
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).
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.
X” cannot generate any solution of Fermat’s last theorem, because we have illogical mathematical deductions, for examples, as follows:
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),
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.
Fermat’s last theorem is simply proved!
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:
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.
Fermat’s last theorem is simply again proved, with the connection to the concept of (X”/Z”)^p ?= 2. Is it interesting?
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