## Mathematics Illuminated

# The Primes

## It is often said that Mathematics is a universal language. No matter one's culture, country, gender, race, or even religion, certain mathematical principles remain true. The fundamental letters of the mathematical alphabet are known as the primes.

Mathematics Illuminates: Unit 1 – The Primes Video Transcript

It is often said that Mathematics is a universal language. No matter one’s culture, country, gender, race, or even religion, certain mathematical principles remain true. The fundamental letters of the mathematical alphabet are known as the primes.

The properties and patterns of prime numbers — whole numbers that are divisible only by themselves and one — have been a source of wonder across cultures for thousands of years, and the study of prime numbers is fundamental to mathematics. This unit explores our fascination with primes, culminating in the million-dollar puzzle of the Riemann Hypothesis, a possible description of the pattern behind the primes, and the use of the primes as the foundation of modern cryptography.

### Unit Goals

- Primes are the fundamental building blocks of arithmetic.
- There are infinitely many prime numbers.
- The fundamental theorem of arithmetic says that each whole number can be uniquely decomposed into products of primes.
- The answer to whether or not there is a pattern behind the primes has eluded mathematicians for millennia.
- One can find arbitrarily long “prime deserts” on the number line.
- One can find finite arithmetic sequences of primes of any length.
- Clock math is a way of generalizing arithmetic.
- Primes and clock math can be used together to create strong encryption schemes.

### Unit Glossary

### ADDITIONAL UNIT RESOURCES: Bibliography

# Bibliography

## Websites

http://primes.utm.edu/howmany.shtml

http://www.rsa.com/

http://www.mersenne.org/

http://www.claymath.org/fas/research_fellows/Green/bio.pdf

http://www.math.ucla.edu/~tao/

http://www.fundinguniverse.com/company-histories/RSA -Security-Inc- Company-History.html

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Ash, Avner and Robert Gross. *Fearless Symmetry: Exposing the Hidden Patterns of Numbers*. Princeton, NJ: Princeton University Press, 2006.

Berlinghoff, William P and Fernando Q. Gouvea. *Math Through the Ages: A Gentle History for Teachers and Others*. Farmington, ME: Oxton House Publishers, 2002.

Bogart, Kenneth, Clifford Stein, and Robert L. Drysdale. *Discrete Mathematics for Computer Science*(Mathematics Across the Curriculum). Emeryville, CA: Key College Press, 2006.

Boyer, Carl B. (revised by Uta C. Merzbach). *A History of Mathematics*, 2^{nd} ed. New York: John Wiley and Sons, 1991.

Burton, David M. *History of Mathematics: An Introduction*, 4^{th} ed. USA : WCB/ McGraw-Hill, 1999.

College of Letters and Science. “Terence Tao: The Mozart of Math.” UCLA. http:// www.college.ucla.edu/news/05/terencetaomath.html (accessed January 25, 2007).

Dantzig, Tobias. *Number: The Language of Science*, The Masterpiece Science Edition. New York: Pi Press, an imprint of Pearson Education, Inc., 2005.

Devlin, Keith. “61: Prime-Time News, ” *Discover*, vol. 26, no. 1 (January 2005).

Drake, Frank and Dava Sobel. *Is Anyone Out There?: The Scientific Search for Extraterrestrial Intelligence.* New York: Delacorte Press, 1992.

Du Sautoy, Marcus. *The Music of the Primes: Searching To Solve the Greatest Mystery in Mathematics*. New York: Harper Collins, 2003.

Du Sautoy, Marcus. 2006. Prime numbers get hitched. *SEED* (March 27),http://seedmagazine.com/news/2006/03/prime_numbers_get_hitched.php (accessed January 19, 2007).

Ellenberg, Jordan. 2006. Math’s architect of Beauty: how Terence Tao’s Quest for Elegance Earned him a Fields Medal and a Macarthur Fellowship. *SEED* (september 22), http://www.seedmagazine.com/news/2006/09/maths_ architect_of_beauty.php (accessed January 25, 2007).

Gross, Benedict and Joe Harris. *The Magic of Numbers*. Upper Saddle River, NJ: Pearson Education, Inc/ Prentice Hall, 2004.

Joseph, George Gheverghese. *Crest of the Peacock: The Non-European Roots of Mathematics*. Princeton, NJ: Princeton University Press, 2000.

Malkevitch, Joe. “Mathematics and Internet Security.” American Mathematical Society. http://www.ams.org/featurecolumn/archive/internet.html (accessed January 19, 2007).

Pfaff, Thomas J. and Max Tran. “The N-Jugs and Water Problem,” *The Pi Mu Epsilon Journal*, vol. 12, no. 1 (Fall 2004).

Rivest, Ron and Robert Silverman “Are ‘Strong’ Primes Needed for RSA?” International Association for Cryptologic Research. http://eprint.iacr. org/2001/007 (accessed 2007).

Rockmore, Daniel.* Stalking the Riemann Hypothesis: The Quest To Find the Hidden Law of Prime Numbers.* New York: Vintage Books, 2005.

Slezeviciene. , R., J. Steuding, and S. Turskien. . “Recent Breakthrough in Primality Testing,” *Nonlinear Analysis: Modeling and Control,* vol. 9, no. 2 (2004).

Singh, Simon. *The Code Book: The Evolution of Secrecy from Mary Queen of Scots to Quantum Cryptography*. New York: Doubleday, 1999.

Tanton, James. “Arithmetic, Algebra and Abstraction,” Text in preparation, to appear 2009.

Wells, David. *Prime Numbers: The Most Mysterious Figures in Mathematics*. Hoboken, NJ: John Wiley and Sons, Inc., 2005.