Professor Simonson History of Math Ingenuity

Fall 1997 MA 149

Write your solutions to these problems as a group. They will be discussed in detail during class. If you cannot solve a problem then report on your group discussion and explain where you got stuck. You will always have the opportunity of rewriting an assignment after class discussion.

1. Euclid Style Proofs

If one defines an "isoceles" triangle as a triangle where two of the sides are equal, then prove that the 2 angles not contained by these sides are also equal. Justify each step in your proof by making reference to the appropriate postulates or common notions in the Euclidean style. This problem just asks to redo the proof we began in class and fill ALL the Euclidean details in your own words.

2. Pythagorean Theorem

a. Using the diagram from class, finish Euclid's proof of the Pythagorean Theorem by drawing in the appropriate line and making the appropriate arguments to show that the remaining larger square equals the area of the larger rectangle. You may argue in a free style without having to mimick a formal Euclidean proof.

b. This diagram is from a Chinese manuscript. Use it to create a proof of the Pythagorean Theorem.

3. Area of a Triangle.

Prove Euclid's Proposition 41 that the area of a triangle is one half the area of any parallelogram which shares the triangle's base and parallels. Make sure to justify all your steps.

4. Pythagorean Triples

A "Pythagorean triple" is a collection of three numbers a, b and c, where a² + b² = c². A "primitive" Pythagorean triple is one where all three numbers have no common factor except 1. For example, (3, 4, 5) is a primitive Pythagorean triple while (6, 8, 10) is a Pythagorean triple but not a primitive one.

Plato suggested that (2n, n²-1, n²+1) is always a Pythagorean triple. Pythagoras suggested that (2n+1, 2n²+2n, 2n²+2n+1) is always a Pythagorean triple. On a stone tablet found in Babylonia, there are etchings of some very large triples, ((120, 119, 169) (3456, 3367, 4825) (4800, 4601, 6649) (13500, 12709, 18451) (72, 65, 97) (360, 319, 481), some of which are shown here in modern notation), which seems to imply that the Babylonians also had some method of constructing them.

a. Prove that all the triples on Plato's and Pythagoras' lists are Pythagorean triples.

b. What triples, if any, appear on both Plato's and Pythagoras' lists?

c. Are the triples generated by these lists primitive? If so, which ones are and which are not?

d. Are there an infinite number of primitive triples? Explain.

e. Do the two methods of Plato and Pythagoras get all the primitive triples?

5. Fermat's Last Theorem

Fermat is the mathematician famous for claiming to have a proof of what is now called "Fermat's Last Theorem". In the margin of his personal copy of Diophantus's Arithmetica, he wrote that it is impossible to divide a cube into a sum of two cubes or in general any nth power into the sum of nth powers where n is greater than 2. Fermat was usually meticulous with his work, so it is a mystery whether he really had a proof of this or not. The theorem was finally proved by Andrew Wiles in 1995, (325 years after Fermat scribbled it down). See figure. Euler, a Swiss mathematician in the 18th century, proved the theorem for the case when n=3 and n=4, although he noted that the techniques he used for each case were so different that he had no hope of unifying the idea to work for a general n.

Show that the more general equation a³ + b³ + c³ = d³ has integer solutions? Conjecture an analagous theorem for fourth powers.