Syllabus Detail

Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 115B: Number Theory

Approved: 2003-05-01 (revised 2013-01-01, G. Kuperberg)
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Elementary Number Theory, 6th Edition by Kenneth H. Rosen; Pearson Publishing; $86.00-121.00.
Search by ISBN on Amazon: 978-0321500311
Prerequisites:
Completion of courses MAT 67 and MAT 115A.
Suggested Schedule:

Lecture(s)

Sections

Comments/Topics

1-5

ch. 7.4, 7.3

Famous functions of number theory: Möbius inversion, multiplicative functions, sum of positive divisors, Morton’s conjecture, perfect numbers

6-13

ch. 9.1-9.3

ch. 10.1, ch. 10.2

More congruences, primitive roots, applications: pseudorandom numbers, the ElGamal Crypto System

14-21

ch 11.1-11.3

Quadratic reciprocity, Legendre symbol, Jacobi symbol, the law of quadratic reciprocity

22-27

ch. 13

Nonlinear Diophantine, equation and continued fractions, Pythagorean triples, Fermat’s last Theorem, Pell’s equation, sums of squares

Additional Notes:
Covers Rosen chapters 7, 9-13
Learning Goals:
The goal is to show students rigorous, beautiful ideas of number theory beyond the most basic, introductory level presented in Math 115A. The ideas of the first quarter are extended. Generally the topic of number theory is advanced from the 17th and early 18th century, to the late 18th century and 19th century. There is also a shift in emphasis from mostly calculation to mostly proofs.

Mastery of this course enhances the students' ability to construct and write proofs; to not only see beautiful ideas of number theory in the time of Gauss, but also reach some of those ideas themselves; and adds to their experience with algebra in general, in particular in association with the Math 150 modern algebra series. AS A CAPSTONE: Students will develop and deepen their understanding of number theory in this second course of the MAT 115 sequence. Through applications to cryptography they will further their understanding of the role number theory plays in the modern world. Students will gain a deeper appreciation of the unity of mathematics by seeing the many different mathematical tools employed to solve number theoretic problems. Via this subject area expertise, they will gain mastery in this area of specialization and improve their ability to communicate mathematics at a capstone level, commensurate with that expected of one with an undergraduate degree in mathematics.
Assessment:
Weekly homework, midterms, and a final exam.