## Fermat's Last Theorem

Today I'm telling my geometry kids a little bit about this most beautiful theorem that I think everyone should at least know of.  Very few people who like math possess the time, talent, and essential obsession to complete a rigorous mathematical proof.  Fewer still get to work on a 300-year-old proof that is famously known as Fermat's Last Theorem.  And only one person, Andrew Wiles, gets to announce to the world in 1993 that he had a proof!

I fell into knowing Fermat's Last Theorem because I'm a math teacher, meaning none of my teachers ever told me about it.  (Why not?!)  I gravitate towards the math corner of every library and bookstore that I go into, often squatting on the floor because I know I'll stay a while.  After reading Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem by Amir D. Aczel, I fell in love with Andrew Wiles.  His incredible brilliance and obsession strangely coexist with his gentle smile and immense modesty.

So it began some 300 years ago that Pierre de Fermat, a French mathematician, was pondering the Pythagoras' equation; most kids learn this a2 + b2 = c2 equation in grade 7.  There are solutions to this equation, called the "Pythagorean Triples," such as: 32 + 42 = 52, and 52 + 122 = 132, and 72 + 242 = 252, and on and on.  I make my geometry kids memorize a few of these.

Fermat probably became bored of the exponent 2 in the Pythagoras' equation after a time, so he entertained the cubed version of the equation, a3 + b3 = c3.  He asked if there were solutions to this equation.  And what about solutions to the general form an + bn = cn where n is greater than 2?

Fermat made a bold claim that no solution existed!  He wrote this claim in the margin of his book, but also added that the margin itself was too small for him to write the proof.  This claim is
Fermat's Last Theorem
.

Professor Andrew Wiles (Cambridge, England) locked himself up for 7 years to focus on proving Fermat's claim.  But Wiles became interested in the problem when he was 10 years old.  What was rooted as a childhood interest, nurtured by zealous passion, inevitably bloomed into an obsession.

I'm no fool to pretend that I understand a fraction of the mathematics in the 100-plus pages that it took for Wiles to compelete the proof.  But it doesn't mean that I can't get goosebumps whenever I share Wiles' story with my students or get emotional whenever I show this video.  (It's a 5-part series.  You can continue to see the rest of the series by clicking on the links from this video.)

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• December 11, 2011 8:34 AM Randy wrote:
Thanks for the Fermat link....I wonder if Fermat took that long to come up w/ a solution and it wouldn't fit in the margin 'cause it was 100 pages long? Doubt it. (that's not the 5th F?)
1. December 11, 2011 12:25 PM Fawn wrote:
Right, Randy, really no one believes that Fermat did as the math that it took in Wiles' proof didn't exist at the time. 5th F is the F-word in my early days of teaching :)
• April 11, 2014 12:44 AM Pham Duc Sinh wrote:
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)

1. Let‘s divide (1) by (Z-X)^p, we shall get:
(X/(Z-X))^p +( Y/(Z-X))^p ?= (Z/(Z-X))^p (2)

2. 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)

3. 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)

4. 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) and (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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