Dr. Sarah’s Worksheet on Andrew Wiles and
Fermat’s Last Theorem

Andrew Wiles

For seven years, Andrew Wiles worked in unprecedented secrecy, struggling to solve Fermat’s Last Theorem, a problem that had perplexed and motivated mathematicians for 300 years. While the statement of Fermat’s Last Theorem itself does not seem important, attempts at solutions inspired many new mathematical ideas and theories, and so in this manner, it is very important. Andrew Wiles’ solution of Fermat’s Last Theorem brought him fame and satisfaction:

*I had this very rare privilege of
being able to pursue in my adult life what had been my childhood dream. I know
its a rare privilege but I know if one can do this it's more rewarding than
anything one can imagine*

Yet, it also brought him pain when a mistake was discovered in his proof. He eventually fixed the mistake and so Fermat’s Last Theorem has finally been proven.

Andrew Wiles describes mathematical research as follows:

*Perhaps I could best describe my
experience of doing mathematics in terms of entering a dark mansion. One goes
into the first room, and it's dark, completely dark. One stumbles around
bumping into the furniture, and gradually, you learn where each piece of
furniture is, and finally, after six months or so, you find the light switch.
You turn it on, and suddenly, it's all illuminated. You can see exactly where
you were.*

We will explore Fermat’s Last Theorem and its solution, but first we will discuss the Pythagorean

Theorem and will look at a proof of it.

**The
Pythagorean Theorem, Its Integer Solutions, and a Proof**

Pythagoras came up with was a mathematical equation that is used all the time in architecture, construction, and measurement. The Pythagorean theorem says that in a right triangle (where one angle equals 90°), the sum of the squares of two sides equals the square of the hypotenuse (the longest side).

In other words, if c is the
hypotenuse, and a and b are the other two sides, then a^{2} + b^{2}
= c^{2}.

An integer solution of an equation is integers a, b and c that satisfy the equation. We all know that

3^{2} + 4^{2} = 9 + 16
= 25 = 5^{2}, and so we see that a=3, b=4 and c=5 is an integer
solution to this equation.
Mathematicians often want to know how many solutions we can find that
satisfy a given equation.
Sometimes there are no solutions, sometimes there are one (3x=6 has only
one integer solution x=2), two (x^{2} -3x+2=0 has only two integer
solutions x=1 and x=2 since 0=x^{2} -3x+3=(x-1)*(x-2)) or many
solutions, and sometimes there are infinitely many solutions. Knowing the number of solutions to an
equation can have important applications.

**Question: **Name an integer
solution to this equation which has a=5.
Why does your solution satisfy the equation? Show your work.

**Question: **We can also show that
a=3n, b=4n and c=5n is an integer solution to a^{2} + b^{2} = c^{2}
for any integer n as follows. (3n)^{2}
+ (4n)^{2} = 3^{2}n^{2} + 4^{2}n^{2} =
9n^{2} + 16n^{2} = (9 +16)n^{2} = (25)n^{2} = 5^{2}n^{2
}= (5n)^{2}. Hence
a=3n, b=4n and c=5n is an integer solution to a^{2} + b^{2} = c^{2}
for any integer n since (3n)^{2} + (4n)^{2} = (5n)^{2}.
How many integer solutions does the equation a^{2} + b^{2} = c^{2}
have? Why?

**Exploration: **In
order to demonstrate why the Pythagorean Theorem holds, think of each side of a
right triangle as also being a side of a square that's attached to the
triangle. The area of a square is
any side multiplied by itself. (For example, the area of a square with side a
is a a = a^{2}.)

To show that a^{2} + b^{2} = c^{2},
follow these steps: At the page
attached on the back of this worksheet, cut out the three squares. Place the squares made from sides a and
b on top of square c. You will have to cut one of the squares to get a perfect
fit. Show Dr. Sarah when you are
finished with this.

**Question: **Explain why your
exploration shows that a^{2} + b^{2} = c^{2}.

** **

** **

**Fermat’s
Last Theorem and Its Proof**

Fermat said that you could not find
any positive whole number solutions to the equation, a^{n} + b^{n}
= c^{n} when n>2. In
other words, there are NO integer solutions to this equation if n>2. In a
mathematical proof you have to write down a line of reasoning demonstrating why
there are no integer solutions. If the proof is rigorous, then nobody can ever
prove it wrong.

**Question:** Why
can’t we just ask a computer to check that there are no solutions?

Andrew Wiles proved Fermat’s Last Theorem is true
by proving that Taniyama-Shimura is true.
At first glance, it appeared that Taniyama-Shimura is unrelated to
Fermat’s Last Theorem.
Taniyama-Shimura said that all elliptic curves (donuts) are modular
forms (symmetries), and gave a dictionary in order to translate problems,
intuition, equations and proofs between these two worlds. Yet, while Taniyama-Shimura had not yet
been proven to be true, many mathematical ideas came to depend on it. Then, the Epsilon-Conjecture came along
which said that *if Fermat’s Last Theorem was false than
Taniyama-Shimura was false*. In other words, if Fermat’s Last
Theorem was false and there was a positive whole number solution to
Fermat’s Last Theorem, then this solution would be so weird that you
could get an elliptic curve that was not modular, and so Taniyama-Shimura would
also be false. The statement *if
Fermat’s Last Theorem is false then Taniyama-Shimura is false* is logically equivalent to the statement *if
Taniyama-Shimura is true then Fermat’s Last Theorem is true*. After
the Epsilon-Conjecture was proven to be true, Andrew Wiles proved
Fermat’s Last Theorem is true by proving that Taniyama-Shimura is true.

In order to explore the idea of the
logical equivalence of statements, let A be the statement *I receive an A in
all my classes this semester*, and B be the
statement *I will receive an A in my math class this semester*. Notice
that the statement *if A then B* is
true because if I receive an A in ALL my classes this semester, then I will
also have to receive an A in my math class. Hence, we also know that the statement *if not B
then not A* is also true since it is
logically equivalent to *if A then B*.

**Question: **Write down the
statement *if not B then not A* and explain why this statement is
true.

**Question:
**Yet, the statement *if B then A*
is not logically equivalent to *if A then B*,
so this statement could be true or false.
Write down the statement *if B then A*. Is the
statement true? Why?

**Question: **Use
only your last answer to explain why the statement *if not A then not B*
is false in this case.

**References**

http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Wiles.html

http://www.pbs.org/wgbh/nova/transcripts/2414proof.html

http://www.pbs.org/wgbh/nova/proof/