Fermat's Enigma

In the early 17th century, French mathematician Pierre de Fermat scribbled a theorem in the margin of a book he was reading. He noted that for the equation

*x*^{n} + y^{n} = z^{n}, if n was greater than 2, then

*x, y,* and

*z* could not all be positive whole numbers. "I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain," he wrote. We'll never know whether Fermat actually proved his proposition, but for 350 years others struggled to prove what became known as Fermat's last theorem. In 1993 Princeton mathematics professor Andrew Wiles finally proved the theorem, in a series of lectures that made headlines worldwide. Read Wiles's saga in

*Fermat's Enigma* by Simon Singh (Walker & Co., 1997). The following puzzles let you dip your toes into the deep waters Wiles charted.

Pythagorean Cutup

Fermat's last theorem is based on the Pythagorean theorem, which states that for any right triangle with side lengths

*x* and

*y* and hypotenuse length

*z, x*^{2} + y^{2} = z^{2}. Geometrically, this means that the areas of the red and blue squares in the diagram at right always add up to the purple square. In this case, 3

^{2} + 4

^{2} = 5

^{2}, or 9 + 16 = 25.

You can verify the formula for this particular 3-4-5 right triangle by cutting up the two smaller squares and rearranging the pieces to make the biggest square, as shown in the diagram at right.

But why limit yourself to squares? You could build similar triangles or other shapes on the sides of the right triangle, as shown below.

Demonstrate the truth of the Pythagorean theorem for this 3-4-5 right triangle by determining how to cut up and rearrange the red and blue shapes below to form the purple shapes. Cut only along grid lines. For each problem, find three different solutions: (a) leave the smaller shape intact and cut the larger shape into three pieces, (b) leave the larger shape intact and cut the smaller shape into three pieces, and (c) cut both shapes into two pieces. Hint: Simple cuts will suffice. Can you find a solution to problem 1(a) that differs from the one shown above?

Triples

The equation

*x*^{2} + y^{2} = z^{2} has many whole-number solutions besides 3

^{2} + 4

^{2} = 5

^{2}. For instance, 6

^{2} + 8

^{2} = 10

^{2} (36 + 64 = 100), 5

^{2} + 12

^{2} = 13

^{2} (25 + 144 = 169), and 8

^{2} + 15

^{2} = 17

^{2} (64 + 225 = 289). Sets of whole numbers that satisfy the Pythagorean formula, like {3, 4, 5} and {6, 8, 10}, are called Pythagorean triples.

**1. **In the Pythagorean triples {3, 4, 5} and {5, 12, 13}, the last two numbers of each set differ by 1. What is the next larger Pythagorean triple in which the last two numbers differ by 1? What is the next such triple after that? There are an infinite number of Pythagorean triples whose last numbers differ by 1, and there is a simple way to prove it. Do you know how? Can you figure out which two consecutive numbers complete the Pythagorean triple {77,

*x, x* + 1}? Hint: Find a pattern in the differences between consecutive square numbers in the diagram at right.

**2. **In the Pythagorean triples {6, 8, 10} and {8, 15, 17}, the last two numbers differ by 2. What is the next larger Pythagorean triple in which the last two numbers differ by 2? What is the next such triple after that? Can you figure out which two numbers that differ by 2 complete the triple {100,

*x, x* + 2}? Hint: Look at the differences between squares of numbers that differ by 2 and find a pattern.

**3.** Can you find all the Pythagorean triples that include 77?

Counterexamples

Pythagorean triples involve square numbers. Cubes won't cut it. Fermat's last theorem states that in the equation

*x*^{n} + y^{n} =

*z*^{n}, there can be no positive whole-number solutions if

*n* is greater than 2, so the equation

*x*^{3} + y^{3} = z^{3} is impossible. Fermat wrote out the proof for

*n* = 4 but only hinted that he had a general proof for all n.

What is odd about Fermat's last theorem is that most variations on it can easily be proved or disproved. The equation

*w*^{3} + x^{3} + y^{3} = z^{3}, for instance, has a solution 3

^{3} + 4

^{3} + 5

^{3} = 6

^{3}. Fermat proved that

*x*^{2} + 2 =

*y*^{3} has only one solution. The equation w

^{4} + x^{4} + y^{4} = z^{4} is harder. In 1988, after 200 years of mathematicians' attempts to prove it impossible, Noam Elkies of Harvard found the counterexample 2,682,440

^{4} + 15,365,639

^{4} + 18,796,760

^{4} = 20,615,673

^{4}.

Can you substitute numbers for the variables

*w, x, y,* and

*z* to make the equations below work? All answers are distinct positive whole numbers between 1 and 10. Hint: Algebra doesn't help.

**1.** *x*^{2} + 2 =

* y*^{3} **2.** *x*^{3} + 1 =

*y*^{2} **3.** *x*^{2} +

*y*^{4} =

*z*^{2} **4.** *x*^{2} +

*y*^{2} =

*y*^{3} **5.** *x*^{3} +

*y*^{3} =

*z*^{3} - 1

**6.** *w*^{3} +

*x*^{3} =

*y*^{3} +

*z*^{3} **Solution** Want to see the

solution to this puzzle?

Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at

www.scottkim.com.

A video of the musical play Fermat's Last Tango is available through

www.claymath.org. And for lovely animated proofs, see the short video The Theorem of Pythagoras, produced by Project Mathematics! and available through the Caltech Bookstore (

www.bookstore.caltech.edu).

© Copyright 2002 The Walt Disney