Instructor

Anna Wienhard

Email

standard email

Reader

Lola Thompson, lola@uchicago.edu

Office hours

Ry360I, Mondays 10:00-11:30 and Wednesdays 17:00-18:00.

Course info

Lectures TThu 10:30 - 11:50, E207

Book

An Introduction to the Theory of Numbers by Niven, Zuckerman and Montgomery

Exams

Hour Test: Thursday, October 26 (in class) Review topics
Final: Tuesday, December 5, 10:30 - 12:30 in E207 Review topics

Due

Homework Assignment

03 Oct 06

Solve Problems 1 (a), 3 (a) (b), 6, 7 on page 17, Problems 20, 23 on page 18, Problems 4, 6, 7 on page 29 and Problems 24, 25 on page 30. Bonus problems: Problem 13 on page 17, Problems 44 and 45 on page 19, Problem 17 on page 29 and show that there are infinitely many primes of the form 6x-1.

10 Oct 06

1) Problems 2,3 5 on p. 56; 16 on p.57; 35 on p.58; 54 on p.59
2) Problem 4 on p.62, Problem 5 (a), (c), (e) on p. 63
3) Let m be a positive integer, m>1. We denote by Z/mZ the set of equivalence classes of integers modulo m. a) Show that Z/mZ is a ring. b) Show that Z/mZ is a field if and only if $m$ is prime. (If you do not know the defintion of a ring or a field, consult Definition 2.12, p.124 and Definition 2.9, p.116)
Bonus: Problem 38 on p. 58. Problem 24 on p. 73, Problem 44 on p. 74.

17 Oct 06

1) Problem 4 on p.72: Find all integers which satisfy the following congruences simultaneously: x=1(mod 3), x=2(mod 4), x=3(mod 5)
2) Problem 10 on p.57
3) Problems 16 and 22 on p. 57
4) Problem 2 on p.82
5) Problem 1 on p.86
Bonus: Prove that any odd integer n>1 is prime if and only if it is not expressible as a sum of 3 or more consecutive positive integers.
Problem 5 on p. 86.

24 Oct 06

1) Problem 1 on p.62, Problem 8 on p.63
2) Problem 15 on p.72, Problem 6, 7 on p.91
3) Problem 10 on p.91
4) Let p>2 be a prime number. Let a/b with a,b relatively prime be the fraction obtained as a/b =1+ 1/2 + 1/3 + ... + 1/(p-1). Show that p divides a.
Bonus: Problem 11 and Problem 12 on p. 63

31 Oct 06

Review material for the midterm
1) Problem 1, 2, 3 on p. 96
2) Problem 1, 5, 7, 13 on p. 106
3) Find a prime p such that 2,3 and 5 are primitive roots modulo p.

07 Nov 06

1) Go through your midterm and think again about the problems you did not solve correctly.
2) Euler's Function: Problem 28, 30, 32, 33, 35 on p. 73
3) Primitive Roots: Problem 2,3,4 on p. 106, Problem 18 on p. 107 (Note that the book sometimes says that a belongs to the exponents h modulo m, which means the same as saying that a is of order h modulo m.)
Bonus: Problem 29 on p. 107

14 Nov 06

1) Problem 22 on p. 107
2) Problem 3,4,7, 9,11 on p. 135/136
3) Problem 1,4 on p. 140
4) Problem 8,11 on p. 141
Bonus: Problem 21, 23 on p. 137

21 Nov 06

1) Problem 2,5 on p. 147, Problem 15 on p. 148.
2) Write down the Farey sequence (between 0 and 1) of order 6 and 7.
3) Problem 1,2,3 page 300, Problem 5 on p. 301.
4) Problem 1, 6 on page 307.
Bonus: Problem 9 on p. 301.

28 Nov 06

1) Problem 1, 3, 4 on p. 327.
2) Problem 2, 3, 4 on p. 333, Problem 1, 2 on p. 336.
3) Problem 2 on p. 344 (you can use Farey fractions or continued fractions to solve this)
Bonus: Problem 6 on p. 341

Practice Final

Practice Final