Math 115 Introduction to Number Theory

Instructor: Chung Pang Mok
Lectures: MWF 12:00 - 1:00 p.m. room 241 Cory
Office hours: Tu 11:00 - 12:00, Th 2:00 - 3:00, room 889 Evans
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.

Mid-term schedule

  • First mid-term: Monday 10/06: section 1.2-1.3; section 2.1-2.3, 2.7, 2.8, up to p.101. Note: You do not need to put too much emphasis on section 2.7 in your review. Questions , Solutions
  • Second mid-term: Monday 11/10: Section 2.8, up to p.101, and section 3.1 - 3.5. Note: for section 2.8, brushing up the key definitions and statements of theorems, would suffice. Sample Problems , Solutions ; Problems on quadratic forms

    Final schedule

  • Friday 12/19, 5:00 - 8:00 p.m. at 60 Evans. Syllabus: materials covered for the first two mid-terms, together with the whole of chapter 7.

    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

    20% homework, 30 % for two mid-terms, 50% final. Homeworks problems will be assigned weekly, usually on Wednesday, 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 09/10: Section 1.2: 1(a), 3(a), 14, 15; Section 1.3: 24, 26 Solutions
  • Homework assignment #2, due on 09/17: Section 2.1: 2, 8, 20, 38; Section 2.2: 5(e) Solutions
  • Homework assignment #3, due on 09/24: Section 2.3: 3, 7, 14, 25, 40 Solutions
  • Homework assignment #4, due on 10/01: Section 2.8: 8, 10, 21, 24 Solutions
  • Homework assignment #5, due on 10/22: Section 3.2: 4(f), 6, 8, 9; section 3.3: 14. Solutions
  • Homework assignment #6, due on 10/29: Section 3.3: 3a, 3d; section 3.5: 1, Extra Questions , Solutions
  • Homework assignment #7, due on 11/05: Section 7.1: 1, 3; Section 7.3: 2, 3b), 3c) Solutions
  • Homework assignment #8, due on 11/26: Section 7.4: 2, 3, section 7.5, 3, 4 Solutions
  • Homework assignment #9, due on 12/10: Section 7.8: 2, 5, 8, 9, 11 Solutions