Biographical information and a portrait al-Khowarizmi (note that there are many variations on his name), as well as facsimiles of some pages of his work, can be seen at the MacTutor History of Mathematics Archive site, at the University of St. Andrews, Scotland.

Other sources for biographical information about al-Khowarizmi are

You can find information about searching algorithms, including the linear (sequential) search algorithm and the binary search algorithm (as well as information about sorting algorithms, including those studied in Chapter 8 of this book) at the following site developed by Thomas Niemann

Section 2.2. The Complexity of Algorithms

Page 110

Information about approximate algorithms for NP problems can be found at

The Prime Pages, written by Chris Caldwell, University of Tennessee, Martin, include a rich collection of information about prime numbers. Among the things you can find on the Prime Pages are various lists of primes, such as the first 100,008 primes, and useful programs for factoring numbers and finding primes. The main site is at

Information about prime numbers and interactive applets for running the sieve of Eratosthenes (see Section 5.6 of the text) and to explore the distribution of prime numbers, twin primes, and the Goldbach conjecture can be found at the Notes and Literature about Prime Numbers site. This site is developed and maintained by Peter Alfeld at the University of Utah.

Biographical information and a portrait of Mersenne can be found at the MacTutor History of Mathematics Archive site, at the University of St. Andrews, Scotland.

At the Great Internet Mersenne Prime Search (GIMPS) site, a wealth of resources concerning Mersenne primes is available, including the current status of the search for new ones, the latest discoveries and news clippings, the history of the discovery of each Mersenne prime, and other information about them. Also, software can be downloaded from this site that can be used to test Mersenne numbers to see whether they are prime. This site also provides a way for someone to join the search by reserving a range of Mersenne numbers to test.

Luther Walsh also has a valuable page on Mersenne primes. You can access this at

A useful discussion of modular arithmetic can be found at the Cut-the-Knot site

Biographical information and a portrait of Karl Friedrich Gauss can be found at the MacTutor History of Mathematics Archive site, at the University of St. Andrews, Scotland.

You can learn more about hashing at

Another place to learn more about hashing functions is in the frequently asked questions (FAQs) of RSA Labs

Page 124

More information about cryptography, including the basics of many different methods for encrypting messages can be found at a number of sites, including

http://www.achiever.com/freehmpg/cryptology/crypto.html (A-Z Cryptology)

http://www.cs.hut.fi/ssh/crypto/intro.html (Introduction to Cryptography)

Section 2.4 Integers and Algorithms

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A tutorial about the Euclidean algorithm and an interactive program that works through the steps of this algorithm can be found at

http://www.maths.monash.edu.au/~apm847j/MAT1130/grant/html/node6.html (Euclidean Algorithm (basic))

Biographical information and a portrait of Euclid can be found at the MacTutor History of Mathematics Archive site, at the University of St. Andrews, Scotland.

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html (Euclid)

Page 130

Information about representations of numbers in different bases and an interactive program for converting between bases can be found at

http://www.maths.monash.edu.au/~apm847j/MAT1130/grant/html/node2.html#SECTION01100000000000000000

Section 2.5 Applications of Number Theory

Page 141

An interesting discussion of the origins, a proof, and examples illustrating the use of the Chinese remainder theorem can be found at

http://www.cut-the-knot.com/blue/chinese.html (Chinese Remainder Theorem)

Historical information about the ancient Chinese origins of the Chinese Remainder Theorem can be found at

http://mathserv.math.sfu.ca/History_of_Math/China/3rdCenturyBC/CRP1.html

Page 145

A biography and a portrait of Pierre de Fermat can be found at the MacTutor History of Mathematics Archive site, at the University of St. Andrews, Scotland.

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html (Fermat) Scotland)

Page 146

More information about the RSA public key cryptosystem and its uses is available at the RSA Data Security site, the company that owns the patents on RSA and commercializes this system.

http://www.rsa.com (RSA Data Security)

In particular, it is worthwhile to consult the Frequently Asked Questions (FAQ) site found in the RSA Laboratories portion of the RSA Data Security site. This is available at

http://www.rsa.com/rsalabs/newfaq/ (RSA Laboratories FAQ on Cryptography)

Section 2.6 Matrices

Page 150

The history of matrices and determinants can be found at the MacTutor History of Mathematics Archive site, at the University of St. Andrews, Scotland.

http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html

Algorithms for efficient multiplication of matrices can be found at the Stony Brook Algorithm Repository

http://www.cs.sunysb.edu/~algorith/files/matrix-multiplication.shtml (1.2.3 Matrix Multiplication)

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