### Math 487, Number Theory

Homework assignments

Instructor:  Paul Jenkins
Office:  320 TMCB, 801-422-5868
Email:  jenkins@math.byu.edu
Lecture:  12:00-12:50 PM MWF, 121 TMCB
Office hours: 1:30-2:30 PM MWF or by appointment
Textbook:  Number Theory, An Introduction via the Distribution of Primes, Benjamin Fine and Gerhard Rosenberger, Birkhäuser, 2007, e-ISBN 978-0-8176-4541-0. The textbook is available for download as a set of pdf files here; if you are not on campus, you will need to use your Route Y login to access the files. From the linked website, you may also order a printed copy for \$25. The book is also published under the ISBN number 978-0817644727.

Prerequisites:  Math 371, Abstract Algebra 1. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical careers, students should be comfortable proving theorems by themselves.

Course Description: This course is aimed at undergraduate mathematics majors. It is a first course in number theory, and is intended to introduce students to number theoretic problems and to different areas of number theory. Number theory has a very long history compared to some other areas of mathematics, and has many applications, especially to coding theory and cryptography.

Minimal Learning Outcomes: Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

1. Divisibility in the integers
• Prime numbers
• Unique factorization
• Euclid’s algorithm
• GCD and LCM
2. Congruence arithmetic
• Complete and reduced residue systems
• Linear congruences
• Chinese remainder theorem
• Polynomial congruences
• Hensel’s lemma
• Legendre and Jacobi symbols
3. Primitive roots
• Existence of primitive roots
• Structure of units modulo nonprimes
4. Number Theoretic Functions
• Moebius Function
• Euler phi function
• Sum of divisors function
• Big O notation
• Little o notation
5. Distribution of Primes
• Definition of Pi(x)
• Estimates of Pi(x)
• Primes in arithmetic progressions
• Bertrand’s Hypothesis
6. Sums of Squares
• Representations of numbers as sums of two and four squares
• Statement of Waring’s Problem

This is a 3 credit class.  The BYU Catalog states that “The expectation for undergraduate courses is three hours of work per week per credit hour for the average student who is appropriately prepared; much more time may be required to achieve excellence.”  Thus, an average student should expect to spend at least 6 hours per week outside of lecture on working problems, reading the textbook, reviewing concepts, and completing assignments.

Grading:  Homework and projects 30%, reading assignments 10%, two midterms 15% each, final exam 30%.
Exams:  Two midterm exams in the testing center on October 3-4 and November 14-15.  The scheduled final exam time is 2:30-5:30 PM on Wednesday December 14.  The final exam will cover all material studied this semester.
Homework:  Homework will be due on Mondays, Wednesdays, and Fridays at 4 PM in the box outside my office door.  Homework assignments will be posted on the course website.  Your homework should be neat and should include enough detail that another student from the class could follow your arguments.  Homework that is not stapled, is excessively sloppy, or is written on paper torn from a spiral notebook may receive less than full credit.  Late homework will not be accepted.  Working in groups on homework is encouraged, but each student should write up each problem, without looking at other students’ written solutions.  The three homework assignments with the lowest scores will be dropped.

Electronic devices:  On exams, only testing center calculators may be used.  Do not use mobile phones or permit them to ring during class.

Preventing Sexual Harassment:  Title IX of the Education Amendments of 1972 prohibits sex discrimination against any participant in an educational program or activity that receives federal funds. The act is intended to eliminate sex discrimination in education. Title IX covers discrimination in programs, admissions, activities, and student-to-student sexual harassment. BYU's policy against sexual harassment extends not only to employees of the university, but to students as well. If you encounter unlawful sexual harassment or gender-based discrimination, please talk to your professor; contact the Equal Employment Office at 422-5895 or 367-5689 (24-hours); or contact the Honor Code Office (4440 WSC) at 801-422-2847.

Students with Disabilities:  Brigham Young University is committed to providing a working and learning atmosphere that reasonably accommodates qualified persons with disabilities. If you have any disability which may impair your ability to complete this course successfully, please contact the Services for Students with Disabilities Office (422-2767). Reasonable academic accommodations are reviewed for all students who have qualified, documented disabilities. Services are coordinated with the student and instructor by the SSD Office. If you need assistance or if you feel you have been unlawfully discriminated against on the basis of disability, you may seek resolution through established grievance policy and procedures by contacting the Equal Employment Office at 422-5895, D-285 ASB.

Honor Code:  In keeping with the principles of the BYU Honor Code, students are expected to be honest in all of their academic work.  Academic honesty means, most fundamentally, that any work you present as your own must in fact be your own work and not that of another.  Violations of this principle may result in a failing grade in the course and additional disciplinary action by the university.  Students are also expected to adhere to the Dress and Grooming Standards.  Adherence demonstrates respect for yourself and others and ensures an effective learning and working environment.  It is the university's expectation, and my own expectation in class, that each student will abide by all Honor Code standards.  Please call the Honor Code Office (4440 WSC) at 801-422-2847 if you have questions about those standards.