M328K (55385): Introduction to Number Theory

Website: https://web.ma.utexas.edu/users/stecker/m328k/

Place & Time: Fall 2021, Tuesday & Thursday 14:00 - 15:30 in PMA 7.124

Instructor: Florian Stecker, PMA 10.132, ut@florianstecker.net

Office hours: Tuesday 15:30 - 16:30 (PMA 10.132) & Thursday 11:00 - 12:00 (on Zoom: 924 6029 3393)

COVID-19: Lectures will be in person again this semester, but attendance is not mandatory. I will upload recordings. Please stay at home if you have symptoms of COVID-19, or any infectious disease. You can submit your homework online or in class. One office hour will be in my office, the other will be virtual. Exams will be in person.

Textbooks: The lecture will be self-contained and you don't need a textbook. If you prefer to learn from a book, or are looking for additional problems, here are some suggestions:

Prerequisites: Math 325K, 333L, or 341 with a grade of at least C-.

Course contents: Number theory is the branch of mathematics which deals with integers, for example trying to answer whether an equation has integer solutions, and how many different ones. This often turns out to be much harder then solving equations in the real numbers!

We will start with basic properties of the integers and proof techniques like mathematical induction. Then we will practice finding and writing proofs while going through the fundamentals of number theory, that is: prime number decomposition, the fundamental theorem of arithmetic, congruences and modular arithmetic, the Chinese remainder theorem, Hensel's lemma, Wilson's theorem, Fermat's little theorem and Euler's generalization of it, multiplicative arithmetic functions, Euler's phi-function. We will also discuss applications of number theory like checksums and public key cryptography. If we have enough time, we might also do one or more of the following topics: primitive roots, quadratic reciprocity, Pell's equation.

You can find information about M328K in general on the math department syllabi page.

Aug 26logic and sets (very low video quality)recording
Aug 31rings, axioms of the integersrecording
Sep 2mathematical induction, divisibility, quotient and remainderrecording
Sep 7GCD, fractions, Bezout's identityrecording
Sep 9Euclidean algorithm, extended Euclidean algorithmrecording
Sep 14prime numbers, Fundamental Theorem of Number Theoryrecording
Sep 16uniqueness in the Fundamental Theorem, irrationality of roots, gcd/lcm with prime factorizationsrecording
Sep 21congruences, Z/mZ is a ringrecording
Sep 23modular exponentiation, linear equations in Z/mZrecording
Sep 28inverses in Z/mZ, Z/pZ is a field, Wilson's Theoremrecording
Sep 30Wilson's Theorem, Fermat's little Theorem, Check digits (no sound after 22 minutes, sorry!)recording
Oct 5ISBN-10 and ISBN-13, some homework problemsrecording
Oct 12Chinese Remainder Theoremrecording
Oct 14polynomial equations, Hensel's Lemmarecording
Oct 19comments on the exam, proof of Hensel's Lemma and examplerecording
Oct 21Euler's Theorem, phi functionrecording
Oct 26ring isomorphisms, summatory function, number and sum of divisorsrecording
Oct 28summatory function of phi, homework problemrecording
Nov 2order, primitive rootsrecording
Nov 4Hensel's Lemma example, Langrange's Theorem, existence of primitive roots mod primesrecording
Nov 9discrete logarithms (recording stopped after 30 minutes, sorry)recording
Nov 11symmetric encryption, Diffie-Hellmanrecording
Nov 16public key cryptography, signaturesrecording
Nov 18finite continued fractionsrecording
Nov 23infinite continued fractionsrecording
Nov 30approximation of irrationals by continued fractions, Pell's equationrecording
Dec 2Pell's equation example, review, discrete logarithm example (some sound problems)recording

Homework: Homework problems will be uploaded to the website every Thursday. Please hand in your solutions the following Thursday, by either giving them to me in class or scanning them and uploading to Canvas by 2 pm.

Your task will usually be to prove some statement. Writing proofs in a clear and understandable way is at least as important as having the right idea. A correct proof that is disorganized, inadequately justified, or just hard to follow for the grader will not receive full credit. I encourage you to discuss your ideas with your classmates, preferably in groups of two, but the solutions have to be written up individually.

Your homework will be graded and returned the following week. You can get up to 10 points on each homework. The total homework score is the sum of these points divided by the maximal possible points, where the two lowest homeworks will be dropped.

Update: as it was requested, I will from now on also publish the LaTeX sources of the homework sheets. To compile them, you need the additional file preamble.tex (which I might update throughout the semester).

Homeworkdue dateremarks
Homework 1 (TeX, solutions)Sep 7
Homework 2 (TeX, solutions)Sep 14
Homework 3 (TeX, solutions)Sep 21
Homework 4 (TeX, solutions)Sep 28
Homework 5 (TeX, solutions)Oct 5
Practice exam (solutions)(none)old midterm from 2019; to be done in 75 minutes
Homework 6 (TeX, solutions)Oct 14fixed a mistake in problem 2 on Oct 8
Homework 7 (TeX, solutions)Oct 21
Homework 8 (TeX, solutions)Oct 28
Homework 9 (TeX, solutions)Nov 4added some hints on Oct 29
Homework 10 (TeX, solutions)Nov 11
Homework 11 (TeX, solutions)Nov 23added clarifications to problem 2 on Nov 19
Homework 12 (TeX)no submission

Exams: There will be on in-class midterm exam and one final exam. The midterm takes place in the usual room, PMA 7.124, but the final is at a different time and takes up to 3 hours.

Grading: The final grade is determined by a weighted average of the homework (20%), the midterm exam (30%) and the final exam (50%). This will be translated into a letter grade by the following table.

90% - 100%A
85% - 90%A-
80% - 85%B+
75% - 80%B
70% - 75%B-
65% - 70%C+
60% - 65%C
55% - 60%C-
50% - 55%D
0% - 50%F

Students with disabilities: If you have an accommodation letter from SSD, send me an email or come talk to me after class. See http://ddce.utexas.edu/disability for more information.