Number Theory and History of Mathematics

University of Redlands, Spring 1997

There is no getting out of it. Through and through the world is infected with quantity. To talk sense is to talk in quantities. . . . You cannot evade quantity. You may fly to poetry and to music, and quantity and number will face you in your rhythms and your octaves. Elegant intellects which despise the theory of quantity are but half developed. They are more to be pitied than blamed.

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

Instructor:

*Meeting Place and Times:* Hentschke 204, Tuesday, Thursday, 8:45 - 11 a.m.

*Instructor Office Hours:* Monday through Friday, 11 a.m. - noon; MWF, 2:30 - 3:30 p.m.; most T/Th afternoons; and by appointment. Check your weekly assignment sheet for T/Th afternoon times and for other changes. Also, I usually am in my office past 3:30 p.m. on Mondays and Fridays; call first to make sure.

*Texts: *

*
Elementary Number Theory *(third edition), by David M. Burton

*
Journey Through Genius: The Great Theorems of Mathematics, *by William Dunham

*Prerequisites:*

MATH 201, Discrete Mathematical Structures; or MATH 204, Problem-Solving Seminar; or MATH 241, Linear Algebra; or permission. MATH 241 strongly recommended.

*Course Objectives:*

To understand several important concepts and techniques from number theory, to include:

polygonal numbers, especially triangular numbers;

divisibility: Division Algorithm, greatest common divisor, least common multiple, Euclidian Algorithm;

primes: Fundamental Theorem of Arithmetic, Sieve of Eratosthenes, infinitude of primes, Goldbach Conjecture;

congruence, linear congruences and the Chinese Remainder Theorem;

Fermat's Little Theorem, Wilson's Theorem, Euler's phi-function, and Euler's generalization of Fermat's Little Theorem;

cryptography: knapsack and RSA public key cryptosystems;

primitive roots, quadratic residues and quadratic reciprocity;

perfect and amicable numbers, Mersenne and Fermat primes;

Diophantine equations, Pythagorean triples and Fermat's Last Theorem.

To understand several historical developments in mathematics, to include:

earliest evidences of counting, development of number systems;

early Egyptian, Mesopotamian, and Chinese mathematics, including origins of the Pythagorean Theorem;

Greek geometry, geometric algebra, and number theory: Thales, Pythagoras, Hippocrates, Eudoxus, Euclid and the *Elements,* Archimedes, Apollonius, Eratosthenes, Heron, Nicomachus, Diophantus, Hypatia;

geometric algebra in the Islamic world: al-Khwarizmi and the House of Wisdom, Umar al-Khayyami;

Chinese, Indian, Islamic and Hebrew origins of combinatorics, Pascal's triangle;

mathematics in the Americas, Africa, and the Pacific Islands;

solution of the cubic equation by 16th Century Italian algebraists;

origins of the calculus: Archimedes, Fermat, Newton, Leibniz;

infinite series: Oresme, Leibniz, Newton, the Bernoullis, Euler;

Fermat's, Euler's, Gauss's and Germain's number theory (see above);

Cantor's infinite cardinals: non-denumerability of the continuum, infinitude of transfinite cardinals, continuum hypothesis.

To improve your ability to read mathematics texts.* *

To improve your ability to prove mathematical theorems.

To improve your ability to solve mathematical problems.

To improve your ability to think logically, analytically, and abstractly.

To improve your ability to communicate mathematics, both orally and in writing. Mathematics 245 satisfies the Writing B (WB) Liberal Arts Foundation requirement for students with junior or senior standing.

*Grading: *Quizzes and examinations, 50%; other coursework, 50%

*Daily Homework Assignments: *There will be a homework assignment, consisting of reading and exercises (from the texts or distributed in class), corresponding to almost every class meeting. You should complete the reading listed for each class meeting *before* the class meeting, and then re-read the material before attempting that day's homework exercises. The exercises are due at the start of the next class period unless I say otherwise in the weekly assignment sheet or in class. Late homework will not be accepted without prior permission. Your three lowest daily homework scores will not be included in your homework average.

You may discuss strategies for solving homework exercises with me and with your classmates, and you should check answers to computational exercises with your classmates. However, unless I say otherwise, the work you hand in must be essentially your own. A good way to ensure this is to write up your solutions independently.

You also may earn homework points for in-class activities and presentations. These assignments cannot be made up, however some of them may count as your one, two, or three lowest daily homework scores and thus not be included in your final homework average.

*Longer Homework Assignments: *There will be several writing assignments throughout the semester. Most of these will be incorporated into the daily homework assignments discussed above, however at least one of these will be a longer research assignment in which you write your own new chapter for the *Journey Through Genius* text. Due dates for topic selection, preliminary reports (3-page pieces of your project), and first and final drafts will be announced in class (soon!).

*Quizzes and Examinations:* There will be a quiz at the start of every class period (except the first), ranging in value from 10 points to 100 points. Quizzes worth more than 10 points will be announced at least one week ahead of time. The quizzes will be comprehensive, but will focus on the most recent reading and exercises. There will be three 100-point quizzes or examinations, to be given on *approximately* March 5, April 9, and May 5. In addition, there will be a comprehensive final quiz or examination, worth up to 200 points. Since our class meeting time spans two class periods, we have a choice of two final examination times: the Office of the Registrar has scheduled the final examination for Friday, May 23, at 6 p.m., or Saturday, May 24, at 3 p.m. As a class, we shall choose and use just one of these final examination times. Quizzes and examinations cannot be made up, however your 50 lowest quiz points will not be included in your final quiz average.

*Time Commitment: * You should expect to spend at least two hours studying outside of class for every hour spent in class. This means that for each of our 2 hour class sessions, you should plan to spend *at least* 4 hours *of quality time* studying outside of class. Of course, if you wish to earn a grade of A or B, you may have to study more!

Course Assignments / Schedule

Great Theorem Research Paper

Biography of a Mathematician Assignment