## Math 471: Theory of Numbers

### T, Th 10:00-11:15 in Goessmann 152

Instructor: Dr. Liubomir Chiriac, 1115J LGRT (email: mylastname [at] math [dot] umass [dot] edu).

Office Hours: Wed 3:30-4:30. You can also set up an appointment by sending me an e-mail.

Course Description: This course is a proof-based introduction to elementary number theory. We will study basic properties of the positive integers including modular arithmetic, linear Diophantine equations, primitive roots, quadratic reciprocity, and Gaussian integers. Several important applications to computer science and cryptography will be discussed.

Textbook:

• Number Theory: A lively introduction with proofs, applications and stories by James Pommersheim, Tim Marks and Erica Flapan.

• Chapters 1-2 offer a quick introduction to the most common methods of proof; I recommend that you read these two chapters on your own. We will cover the core material in Chapters 3-9, as well as some more advanced topics such as: Chapter 10 (Primitive Roots), Chapter 11 (Quadratic Residues) and Chapter 13 (Gaussian Integers).

• Homework and participation - 30%;

• Two Midterms - 20% (each);

• Final - 30%.

90-100

85-89

81-84

77-80

73-76

69-72

65-68

61-64

56-60

51-55

0-50

A

A-

B+

B

B-

C+

C

C-

D+

D

F

Exams:

• Midterm 1 - Thursday, 10/11/18, in-class. Sections covered: 3.1-3.5, 4.1-4.2, 5.1-5.4, 6.1-6.2, 7.1-7.2, 7.4.

• Midterm 2 - Thursday, 11/15/18, in-class. Sections covered: 8.1-8.5, 9.1-9.5, 10.1-10.3.

• Final - Friday, 12/14/18, 10:30 am-12:30 pm, in LGRC A203. The focus of the exam will be on Chapters 10,11 and 13.

Here are some practice problems for the Final.

• Homework:

• Homework will be assigned every Thursday and will be due at the beginning of class the following Thursday.

• Assignments turned in by the next Tuesday will receive 75% credit. No assignments will be accepted after that.

• Note that you will need to do the reading to be able to do the homework. If you're stuck, it's ok to consult your classmates or me. However, the work you turn in must be entirely your own, in your own words!

• All Exercises are from the textbook. Note that "x.y:N" is to be read Exercise N from Section x.y.

• Problem Set 1 (Due 09/13): 2.2:8 3.1:27 3.2:4,15 3.3:7 3.4:6,10,11,25 3.5:15

Problem Set 2 (Due 09/20): 3.5:29,30 4.2:3,4,5(attach your program to the HW),7,9,10 5.1:7,11

Problem Set 3 (Due 09/27): 5.2:11 5.3:11,18 5.4:11,12 6.1:9 6.2:1,3,12,22

Problem Set 4 (Due 10/4): 7.1:9 7.2:7,8,9 7.4:4,5,12,14,21,29

Problem Set 5 (Due 10/11): Practice Midterm I

Problem Set 6 (Due 10/18): 8.2:1,5,8,9,18 8.3:4,5,7,8,10

Problem Set 7 (Due 10/25): 8.4:2 8.5:1,3 9.1:10,11,13 9.2:3,5,9,18

Problem Set 8 (Due 11/1): 9.3:9,12,15,16 9.4:1,2 9.5:7,9 10.1:11,14

Problem Set 9 (Due 11/8): 10.1:31 10.2:2,9,15 10.3:3,4,7,11,15,17(b)

Problem Set 10 (Due 11/15): Practice Midterm II

Problem Set 11 (Due 11/29): 11.1:3 11.2:2,10 11.4:11,14,17,18,19 11.5:8,12

Problem Set 12 (Due 12/6, Last HW!): 13.1:8,10,17 13.2:2,4,10,19 13.4:2 13.5:5 13.6:6.

Attendance: Attendance is required during lectures. I consider attendance AND participation important ingredients for your success in the course. Frequent absences will be reflected in your grade.

Computers: Number theory is an inherently computational subject and we will undoubtedly have some use for machine calculations over the course of the semester. When the time comes, you will probably be able to get away with Maple or Mathematica if you're used to those packages. However, there is a far better option out there: Pari/GP. Pari/GP has two main advantages: 1) it is specifically designed for use in number theory; and 2) it is available at a very reasonable price: free. We'll deal with this when the time comes, but if you're curious, everything you ever wanted to know about Pari/GP can be learned at The PARI homepage.

Make-up Exam Policy: If you have a legitimate (e.g. multiple exams at the same exact time, medical problems, emergency absences, religious observances) scheduling conflict with any of the exams, it is your responsibility to notify me of the conflict as soon as you become aware of it. Please note that previously arranged travel plans are not a valid reason to be given a make-up exam.