M 328K Introduction to Number Theory: Fall 2017

Day/Time: TTh 9:30am-11am; Location: SZB 284; Unique: 54212

Instructor
Mirela Ciperiani (mirela at math dot utexas dot edu); Office: RLM 12.164

Office Hours
Tuesday 2:30-4:30pm in RLM 12.164.

Text
Elementary Number Theory and it applications by Kenneth H. Rosen, Sixth Edition. The book should be available at the University Co-op.

Objectives of the Course
This course aims to be a little window into the beauty of number theory as well as a tool to sharpen the clarity of your thinking in proof writing.

Prerequisites
Mathematics 325 or 341 with a grade of at least C.

Grader
Huyen Pham

Exams
Midterm Exam I: Tuesday, October 17, 9:30-11am in SZB 284
Midterm Exam II: Thursday, November 16, 9:30-11am in SZB 284
Final Exam: Monday, December 18, 2:00-5:00 pm in SZB 284
  • You must bring a valid photo ID to all exams.
  • Notes, books, computers, phones, and calculators cannot be used or even visible during exams.
  • Your final exam grade will replace the lower of your two midterm exam grades, if the final exam score is higher than either of them.
  • Students will be excused from the exams only because of a serious illness or another emergency of similar gravity. In such a case you must contact me via email before the exam (if physically possible) and have documentation indicating your inability to take the exam at the scheduled time. In such a case the grade weight of a midterm will be shifted to the following exams in the most advantageous way for the student and a make-up final will be given.

Homework
Most of the homework problems will consist of statements to be proved, and a few will involve computations. With respect to the problems that are statements to be proved, a correct solution consists of a complete proof of the statement. Proofs should be written out in a manner similar to the proofs in the text. Students are encouraged to discuss the homework with others but should write their solutions individually.
  • The homework assignments will be posted on Canvas Assignments by Friday morning each week.
  • Homework is due on Thursdays at the beginning of class and will be returned to you on the following Tuesday.
  • No late work will be accepted. We will drop two of the assignment scores to allow for legitimate reasons for not turning in or underperforming in an assignment.

Grading
Homework 20%
Midterms 25% (each)
Final exam 30%
  • Plus/minus grades will be assigned for the final grade in this course.
  • All your grades will be posted on Canvas. It is your responsibility to insure that your grades are recorded correctly on Canvas. If an error occurs you must alert the instructor within two weeks from the date when the correct grade should have appeared.
  • On all work, your grade will be computed as a percentage: the number of points you earned divided by the number of points possible. The percentages of each type of work that will be used to compute your final grade are given above. Your letter grade will be given based on your numerical average earned in the class, on a scale not stricter than the following: you are guaranteed a D for 40 or above, C- for 50 or above, C for 55 or above, C+ for 65 or above, B- for 70 or above, B for 75 or above, B+ for 85 or above, A- for 90 or above, and an A for 93 or above.

Conflicts
Students with special concerns, be they athletes who might miss class meetings, students with religious observances that interfere with class meetings, or students with disabilities who need special accommodation, are all supposed to notify the instructor by September 15th about these special needs.

Disabilities
Students with disabilities may request appropriate academic accommodations from the Division of Diversity and Community Engagement, Services for Students with Disabilities, 512-471-6259.

Deadlines for Dropping a Course
If you drop a class on or before September 15, the class will not show up on your transcript. If you drop a class after that date, the course will show up on the transcript with a "Q'' grade. After November 7, it is not possible to drop a course except for extenuating (usually non-academic) circumstances.

Student Conduct
  • Attendance: This course is structured with the expectation that you will attend every lecture, and your grade will benefit from your attendance. Of course, sometimes an absence is necessary. In such a situation, you should contact a classmate to get notes, due dates and other information for the class you missed. Please introduce yourself to and write the contact information of at least two classmates.

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  • Please come to class on time. If you are late or need to leave early for some legitimate reason, please sit near the exit. Coming and going during class is distracting to your fellow students. Please do not talk or otherwise disturb students in the class who are trying to learn.
  • All computers, cell phones and other electronic devices must be put away out of sight during class and during exams.
  • Academic Honesty: Copying your written work from somebody else or from any other source is considered cheating and will be dealt with severely. Keep in mind that most students are honest. Honest students do not like cheating, and they do report what they see. Cheating will be penalized as harshly as possible under the rules of UT.




Schedule of lectures

This schedule is tentative and may be modified as necessary.

Date
Reading
  Aug. 31   1.1-1.2: Numbers and Sequences, Sums and Products
  Sept. 5, 7   1.3-1.5: Induction, Fibonacci numbers, Divisibility
  Sept. 12, 14   3.1-3.4: Primes and their distribution, Greatest common divisors, The Euclidean algorithm
  Sept. 19, 21   3.5-3.7: The Fundamental theorem of arithmetic, Factorization, Linear Diophantine problems
  Sept. 26, 28   4.1-4.3: Congruences, The Chinese Remainder theorem
  Oct. 3, 5   4.4-5.1: Solving polynomial congruences, Systems of linear congruences, Divisibility tests
  Oct. 10   6.1: Wilson's theorem, Fermat's little theorem
  Oct. 12
Review
  Oct. 17
First Midterm
On the material covered Aug. 31 - Sept. 28
  Oct. 19   6.2-6.3: Euler's theorem, Pseudoprimes
  Oct. 24, 26   7.1-7.4: Multiplicative functions
  Oct. 31, Nov. 2   9.1-9.3: Primitive roots
  Nov. 7, 9   11.1-11.2: Quadratic residues and nonresidues, The law of quadratic reciprocity
  Nov. 14
Review
  Nov. 16
Second Midterm
Focussed on the material covered Oct. 3 - Nov. 2
  Nov 21   13.1-13.2: Pythagorean triples, Fermat's last theorem
  Nov. 28, 30   13.3- 13.4: Sums of squares, Pell's equation
  Dec. 5, 7   14: Gaussian integers