# Mathematics 115

## Fall, 2012 TuTh 12:40-2PM, 4 Evans Hall

 Professor Kenneth A. Ribet email: Telephone: 510 642 0648 Fax: (510) 642-8204 Office hours (885 Evans Hall)

## Overview

This course is elementary in the sense that no specific course is needed as a prerequisite. On the other hand, the ability to read and write proofs is pretty much essential. Math 55 and Math 113 provide helpful background. As we encounter objects that can be regarded as groups, rings, fields, homomorphisms,..., I will mention the connection to Math 113 in class.

## Textbook

An Introduction to the Theory of Numbers, Fifth Edition by Ivan Niven, H. S. Zuckerman and Hugh L. Montomery. Although the current edition was published 20 years ago, this book remains one of the definitive introductions to the subject. It is renowned for its interesting, and sometimes challenging, problems. Please see Montgomery's home page for the book and especially his lists of typos and errors in the book. Note that there are multiple lists because the book has been reprinted several times.

This book is not cheap, but it should be easy to find used copies: “Niven & Zuckerman” (as the book is widely known) has been used repeatedly at Berkeley.

#### Chapter-by-Chapter Description

• Chapter 1: Divisibility, the Euclidean algorithm, primes and the Fundamental Theorem of Arithmetic, proofs of the infinitude of primes, the binomial theorem. Much of this material will be familiar from Math 55, but I hope that the book and my lectures will provide additional perspective. We can also do some computations with sage (see below).
• Chapter 2: Congruences, Euler's phi function, the ring of integers mod m, Fermat's little theorem and its generalization by Euler, Wilson's theorem, Fermat's theorem characterizing integers that are sums of two squares, polynomial equations mod m, the Chinese remainder theorem, Pollard's rho method for factoring, RSA cryptography, Hensel's lemma, primitive roots, quadratic residues and higher residues. This chapter ends with some material on groups, rings and fields. We will not discuss this material in any depth in class, but I will allude to it from time to time, as I explained in connection with Math 113.
• Chapter 3: Quadratic reciprocity, the Jacobi symbol and applications, binary quadratic forms. I hope to give several proofs of quadratic reciprocity, starting with the proof in the book. This means, in particular, that I will be lecturing on material not in the book. Similarly, when speaking about binary quadratic forms, I hope to explain a new way of looking at the classical results that was discovered recently by Manjul Bhargava.
• Chapter 4: Some functions of number theory. We will discuss only some of the material in this chapter. For example, we will prove the Moebius inversion formula.
• Chapter 7: We will discuss continued fractions, as time allows. For example, we will see in the beginning of the chapter that the Euclidean algorithm of chapter 1 is a method for finding the continued fraction expansion of a rational number.

## Sage

Sage is a free open-source mathematics software system that does number theory calculations that will illustrate and illuminate the material of the course. Even before the semester begins, you can become familiar with sage by taking the tour and then experimenting with the software. When I taught Math 116 last semester, I projected a sage notebook from my laptop for a substantial fraction of each lecture period. Don't be surprised if I continue in that direction this semester. I will try to assign interesting homework problems that require sage for calculations.

You can download the software for your Windows, Linux or MacOS X box. Alternatively, you can run sage online at http://www.sagenb.org/ after you create an account for yourself.

## Examinations

Please do not plan travel on the dates of these exams. If you believe that you have a conflicting obligation because of an intercollegiate sport or other extracurricular activity, please read these guidelines immediately.

For practice exams, you might consult the web pages for my previous Math 115 courses

and for a recent course by Martin Olsson. You may also consult Richard Borcherds's Math 115 page for Fall, 2003.

Course grades will be based on a composite numerical score that is intended to weight the course components roughly as follows: midterm exams 15% each, homework 25%, final exam 45%.

When I taught this course in 2011, there were 33 registered students. Grades were distributed as follows: 11 As, 14 Bs, 4 Cs, 4 D/F. This rough distribution ignores +'s and -'s. Some students took the course P/NP. Their letter grades were converted to P or NP when I entered the final grades.

For this course (Fall, 2012), there were 36 registered students. They received 17 As, 13 Bs, 3Cs and 3Fs. The students with the a P/NP grading option had their grades converted when final grades were entered.

You can look at the course evaluations that my Math 115 students wrote in December, 2011 as well as the evaluations for this course. I encourage suggestions and comments about my teaching style and the evolution of the course. You can make them anonymously in various ways (e.g., by slipping a note under my office door) or just present them in email or a face-to-face conversation.

## Homework

1. Assignment due August 30, 2012: §1.2, problems 1, 2 and 3: all parts, using sage; also problems 4b, 5, 7, 15, 25, 27, 28, 47
2. Assignment due September 6, 2012: §1.3, problems 2, 8, 10, 11, 13, 16, 17, 26, 28, 31
3. Assignment due September 13, 2012:
• §1.3, problems 42, 44, 48 (for the last problem, see the first lines of the basic properties section of the wikipedia Fermat number entry)
• §1.4, problems 3, 4
• §2.1, problems 6, 13, 26, 30 (check using sage), 34, 35, 36, 37, 43
4. Assignment due September 20, 2012:
• §1.2, problem 50
• §1.3, problems 27, 29, 36
• §1.4: Let n = 5k + j with k at least 1 and j = 0, 1, 2, 3 or 4. Show that k is congruent mod 5 to the binomial coefficient "n choose 5".
• §2.1, problems 33, 40
• §2.2, problems 8, 9
5. Assignment due September 29, 2012: §2.3, problems 4, 8, 13, 14, 17, 18, 26, 27, 29, 30, 39
6. Assignment due October 4, 2012:
• §2.7, problems 1 (using sage if possible), 6, 12
• §2.8, problems 3, 12, 16, 18, 20, 23
7. Assignment due October 11, 2012:
• §2.8, problems 24, 25, 29, 30, 31
• §3.1, problems 4 (just use sage), 5 (use sage to compute), 6 (use sage to avoid tedium)
8. Assignment due October 18, 2012:
• §3.1, problems 13, 14, 15, 16, 17, 18
• §3.2, problems 6, 7, 8, 11, 13
9. Assignment due October 25, 2012:
• §2.3, problem 37
• §2.7, problems 13, 14
• §2.8, problems 33, 34, 35
• §3.3, problems 14, 15
10. Assignment due November 1, 2012:
• §4.1, problems 2, 5, 9, 14, 17, 19, 34
• §4.2, problems 10, 12, 16, 19
11. Assignment due November 8, 2012
12. Assignment due November 15, 2012:
• §10.1, problems 3, 4
• §10.2, problems 1, 2, 4, 5, 6
• §10.3, problem 3
• §10,4, problem 2
13. Assignment due November 27, 2012:
• §5.4, problems 14
• §5.6, problems 1, 9, 10 (skipping the part about "nonsingular")
• §5.7, problems 9, 10
14. Assignment due December 6, 2012:
• §2.4, problems 16, 17, 18
• §5.8, problem 4
• Find the order of the point P = (94269158776925 , 1102841572571055) on the elliptic curve defined by y^2 = x^3 + 183821385707290x + 1153449657807210 over the ring of integers modulo M=1636998688431221. Do this by using the elliptic curve factoring method to factor M; then use sage to compute the order of P modulo each of the factors of M.

## Calendar

The calendar that follows (at least if you're logged into gmail!) attempts to call your attention to "events" of interest to math 115 students: class meetings, office hours, exams, optional class get-togethers (coffee, lunch, breakfast), and special lectures for undergraduates.

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