## Introduction to Number Theory, Math 104, Fall quarter 2005:

Course material: We use the book `Number theory with computer applications' by Kumanduri and Romero. No programming will be required. A tentative outline of the material to be covered is given below. This may change during the course, or we may move faster or slower than projected:

First quarter: chapters 1-5, 7, 9,

Second quarter: chapters 11, 14, 17, 19 possibly more material on elliptic curves, possibly chapter 15,

Office hours: M: 2-3 W: 1:30-2:30 and by appointment

Office: APM 5256, tel. 534-2734

Teaching assistant: Amanda Beeson, APM 2301 office hours: M: 1-2 APM 2301, Wed: 6-7 Solis 110,

Computation of grade: Final 50%, Midterm 30%, Homework 20%

Dates of quizzes and midterm:

midterm: 11/9 or possibly 11/11

final: 12/8: 11:30-2:30

Homework assignments: The exam problems will be similar to homework problems. So doing the homework problems is part of your preparation for exams, and is far more important than the 20% what they count for the grade. Homework is to be turned in on Mondays. There is a box in front of APM 2301, or you can turn it in in class.

for 9/28: p13: 2, 8, 16, 27, p24: 6

for 10/10: p 31: 5, 13, p. 45: 1b, 4, 8, 15, 23, p. 53: 1ab, 4, 8, p. 67: 2, 6, 9,

for 10/17: p72: 2, 6, 9, p80: 1a, 2a, 5, p. 86: 1(a) (use Example 3.4.7) (b), 2(a), 3,

for 10/24: p86: 5, 8(enough if you do it for m odd), p. 96: 2cd, 6 (If you got stuck in 6, do 4a instead), p. 107: 1c, 2, 8,

for 10/31: p.110: 1, 11, p. 116: 1ab, 3 (extra credit: Let m,n be coprime. Show that a^d congruent 1 mod mn for d= lcm(\Phi(m),\Phi(n)) if a is coprime to mn). p. 119: 1, 5 (for 5c: observe that f(p)=(p-1)!=a_0; one also needs to assume p>3).

for 11/7: p. 143: 2, 7, p. 149: 1, 9,

for 11/21: p. 175: 4, 5(a), 8, 13, p. 181: 1ab, 7, 10

for 11/30: p. 182: 11a (see Example 7.2.13), 17 (you may use that any Carmichael number n is squarefree, i.e. it is a product of distinct prime numbers; hint: find an a whose order is divisible by p-1 for any p|n). p. 185: 2(b) (find a primitive root, and use Shank's algorithm following Example 7.3.6); also solve for k in 17k^5 congruent 29 mod 37 (k^5 means the 5th power of k), p. 174: 1d, p. 191: 1a

Final The final will take place on Thursday, 12/8, 11:30-2:30 in class. You will be allowed to use one hand-written cheat sheet and a calculator. Here are some problems from an old final:

Other possible problems, besides the ones done in class, might involve discrete logarithm problems, calculating the order of a residue class or other problems similar to given homework problems.

office hours: (for exam week): T3-5, W2-3

Midterm The midterm will take place on Wednesday, 11/9, in class. The material will go until the last homework assignment, due 11/7. You are allowed to use one hand-written cheat sheet, but no calculators or books. The problems will be similar to homework problems. Below you can find problems from previous midterms I have given. While this may give you some ideas about what problems to expect, it does not mean that the up-coming midterm will be of the exactly same form.