Dr. Sarah’s Worksheet on Andrew Wiles and Fermat’s Last Theorem
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 a2 + b2 = c2.
An integer solution of an equation is integers a, b and c that satisfy the equation. We all know that
32 + 42 = 9 + 16 = 25 = 52, 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 (x2 -3x+2=0 has only two integer solutions x=1 and x=2 since 0=x2 -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 a2 + b2 = c2 for any integer n as follows. (3n)2 + (4n)2 = 32n2 + 42n2 = 9n2 + 16n2 = (9 +16)n2 = (25)n2 = 52n2 = (5n)2. Hence a=3n, b=4n and c=5n is an integer solution to a2 + b2 = c2 for any integer n since (3n)2 + (4n)2 = (5n)2. How many integer solutions does the equation a2 + b2 = c2 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 = a2.)
To show that a2 + b2 = c2, 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 a2 + b2 = c2.
Fermat’s Last Theorem and Its Proof
Fermat said that you could not find any positive whole number solutions to the equation, an + bn = cn 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.