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

Textbook

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

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

• First midterm exam, Thursday, September 20, 2012, in class (questions and skeletal solutions, mean 19.14, standard deviation 7.29).
• Last midterm exam, Thursday, October 25, 2012, in class (questions and skeletal solutions, mean 19.79, standard deviation 7.55).
• Final examination, Friday, December 14, 2012, 8-11AM in 3113 Etcheverry Hall (questions and skeletal solutions). On this 54-point exam, the mean was 34.26 while the standard deviation was 11.84.

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

• Spring, 1998
• Fall, 1999
• Fall, 2000
• Fall, 2006
• Fall, 2011

Grading

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
   • §2.9, problems 1ad, 7
   • §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.

Some web resources related to the course

• What is the smallest prime? by Chris K. Caldwell and Yeng Xiong.
• Reinventing the Classroom by Harry R. Lewis. Should we organize Math 115 this way?
• The on-line encyclopedia of integer sequences
• The prime pages
• Eckford Cohen's article about the prime factorization of n!, which includes a weird proof that there are infinitely many primes. (You have access to the full article when authenticated as coming from Berkeley.)
• A compendium of proofs of Fermat's 1654 theorem to the effect that primes that are 1 mod 4 are sums of two squares. See especially Don Zagier's proof of the theorem. See also the PlanetMath account of the proof, which includes a simple proof of the uniqueness.
• While we're at it, you might enjoy Don Zagier's article The first 50 million prime numbers
• A fairly old list of alternative references for Math 115.
• An interesting and fairly recent discussion about the question of whether ((p-1)/2)! is +1 or -1 mod p when p is 3 mod 4. This residue class is 1 for p = 3, 23, 31, 59, 71 and 83 but -1 when p is 7, 11, 19, 43, 47, 67, 79. It's hard to see a pattern, isn't it?
• An alternative proof of the law of quadratic reciprocity. If you prefer, you can consult the original 1872 manuscript by Zolotarev. In fact, you may find this mathoverflow discussion to be of great interest. One manuscript that mathoverflow refers to is this shortened proof by Wouter Castryck.
• Just to see if anyone is reading these links, here's an article about a lecture that I gave on October 16, 2011 at Humboldt State University. I gave two more lectures the next day.
• Wikipedia's discussion of the Lucas-Lehmer test. This web page was the basis for my lecture on November 3, 2011.
• William Stein's book Elementary Number Theory: Primes, Congruences, and Secrets.
• Henry Cohen's article A Short Proof of the Simple Continued Fraction Expansion of e.
• Hendrik Lenstra's 2002 article on Pell's equation. See also The Number of Real Quadratic Fields Having Units of Negative Norm by Peter Stevenhagen.

Calendar

Last Updated: