Project 4.1

What do the following people have in common?
Corazon Aquino, former President of the Philippines
Leon Trotsky, revolutionary
Carole King, singer-songwriter
Heloise (Poncé Cruse Evans), columnist, Hints from Heloise
Florence Nightingale, pioneer in professional nursing

Project 4.2

Project 4.3

Project 4.4

Project 4.5

Project 4.6

Investigate some of the properties of primes not discussed in the text. Why are primes important to mathematicians? Why are primes important in mathematics? What are some of the important theorems concerning primes?

Reference:

Martin Gardner, "The Remarkable Lore of Prime Numbers," Scientific American, March 1964.

Project 4.7

Project 4.8

Project 4.9

We mentioned that the Egyptians wrote their fractions as sums of unit fractions. Show that every positive fraction less than 1 can be written as a sum of unit fractions.

References:

Bernhardt Wohlgemuth, "Egyptian Fractions," Journal of Recreational Math, Vol. 5, No. 1 (1972), pp. 55-58.

Project 4.10

The Egyptians had a very elaborate and well developed system for working with fractions. Write a paper on Egyptian fractions.

References:

George Berzsenyi, "Egyptian Fractions," Quantum, November/December 1994, p. 45. Follow up comment in The College Mathematics Journal, March 1995, p. 165.
Richard Gillings, Mathematics in the Time of the Pharaohs (New York: Dover Publications, 1982).
Spencer Hurd, "Egyptian Fractions: Ahmes to Fibonacci to Today," Mathematics Teacher, October 1991, pp. 561-568.

Project 4.11

Project 4.12

Prove that the positive square root of 2 is not rational.

Project 4.13

Write a paper or prepare an exhibit illustrating the Pythagorean theorem. Here are some questions you might consider:
What is the history of the Pythagorean theorem?
What are some unusual proofs of the Pythagorean theorem?
What are some of the unusual relationships that exist among Pythagorean numbers?
What models can be made to visualize the Pythagorean theorem?

Project 4.14

Symmetries of a Cube
Consider a cube labeled as shown below:

List all the possible symmetries of this cube. See Problem 60 in, Problem Set 4.6 to help you get started.

Project 4.15

What is a Diophantine equation?

Reference:

Warren J. Himmelberger, "Puzzle Problems and Diophantine Equations," The Mathematics Teacher, February 1973, 136-138, or a more complete reference, see any number theory textbook.

Project 4.16

Prepare an exhibit on cryptography. Include devices or charts for writing and deciphering codes, coded messages, and illustrations of famous codes from history. For example, codes are found in literature in Before the Curtain Falls, by J. Rives Childs; The Gold Bug, by Edgar Allan Poe, Voyage to the Center of Earth, by Jules Verne.

References:

Andree, Richard V., "Cryptography as a Branch of Mathematics," The Mathematics Teacher, November 1952.
Gardner, Martin, "Mathematical Games Department," Scientific American, August 1972, pp. 114-118.
Shasta, Dennis, Codes, Puzzles, and Conspiracy, Menlo Park, CA: Dale Seymour Publications, 1993.

Project 4.17

Write a paper on the importance of cryptography for the internet. You might begin with the Scientific American article by Philip Zimmerman and conclude with the RSA "Secret-Key Challenge."

References:

Zimmerman, Philip, "Cryptography for the Internet," Scientific American, October, 1998, pp. 110-115.

RSA Data Security Secret-Key Challenge:
http://www.rsasecurity.com/

Note: If you look at the page Status and Prizes, you will see the contest RC5-32/12/5 This $10,000 prize was won by deciphering the following secret message: "Strong cryptography makes the world a better place."