Math 115 Introduction to Number Theory

Instructor: Chung Pang Mok
Lectures: MW 4:00 - 5:30 p.m. room 4 Evans
Office hours: T 2:30 - 3:30, Th 11:00 - 12:00
Email address: mok@math.berkeley.edu

About the course

Pierre de Fermat is generally regarded as the father of modern number theory. In his time, he stated interesting results on prime numbers represented by quadratic forms, mostly without proofs. Later mathematicians, most notablty, Euler and Lagrange, attempted to reconstruct Fermat's arguments. In doing so, Fermat's results were fitted into a coherent picture. In 1801, the German mathematician Gauss publised his work Disquisitiones Arithmeticae (Research in Arithmetic) , which laid the foundations of modern number theory, and sources of much further research.

A central result in the subject is the law of quadratic reciprocity . Gauss realised its importance, calling it the jewel of arithmetic (and gave eight proofs of it during his lifetime). In this course, we are going to understand the development of the subject.

Topics to be covered

  • Divisibility
  • Congruences
  • Quadratic Reciprocity and Quadratic Forms
  • Simple Contined Fractions

    Texts for the course

    The official text is

  • Niven, Zuckerman and Montgomery, An Introduction to the Theory of Numbers, Wiley, 5th edition.

    from where most of the homework problems will be taken.

    Number theory is blessed with many excellent introductory texts. Some of which I recommend as supplementary references for the course are:

  • H. Davenport, The Higher Arithmetic, Hutchinson, London.
  • G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Clarendon Press, Oxford.
  • L. K. Hua, Introduction to number theory, Springer.

    They are available in the Math and Stat library on the first floor of Evans, and also in the Moffitt undergraduate library.

    Grading Policy

    15% + 30 % for two mid-terms + 55% final. Homeworks problems will be assigned weekly, and due the following week. Check this page for the assignments.
    Collaboration is encouraged, but you have to write the final solution on your own.

    Assignments

  • Homework assignment #1, due on 09/09: Section 1.2: 1(a), 3(a), 14, 15; Section 1.3: 24, 26 Solutions
  • Homework assignment #2, due on 09/16: Section 2.1: 2, 8, 20, 38; Section 2.2: 5(e) Solutions
  • Homework assignment #3, due on 09/23: Section 2.3: 3, 7, 14, 25, 40 Solutions
  • Homework assignment #4, due on 09/30: Section 2.8: 8, 10, 21, 24 Solutions
  • Homework assignment #5, due on 10/07: Section 3.2: 4(f), 6, 8, 9; section 3.3: 14 Solutions
  • Homework assignment #6, due on 10/28: Section 3.3: 3a, 3d; section 3.5: 1, Extra Questions , Solutions
  • Homework assignment #7 : Section 7.1: 1, 3; Section 7.3: 2, 3b), 3c) Solutions