## M328K (55385): Introduction to Number Theory

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:

• Elementary Number Theory and Its Applications by Kenneth H. Rosen
• Elementary Number Theory: Primes, Congruences, and Secrets by William Stein (available for free here)
• Number Theory and Geometry by Álvaro Lozano-Robledo (partially available for free through the UT library)

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.

Schedule/recordings:
 Aug 26 logic and sets (very low video quality) recording Aug 31 rings, axioms of the integers recording Sep 2 mathematical induction, divisibility, quotient and remainder recording Sep 7 GCD, fractions, Bezout's identity recording Sep 9 Euclidean algorithm, extended Euclidean algorithm recording Sep 14 prime numbers, Fundamental Theorem of Number Theory recording Sep 16 uniqueness in the Fundamental Theorem, irrationality of roots, gcd/lcm with prime factorizations recording Sep 21 congruences, Z/mZ is a ring recording Sep 23 modular exponentiation, linear equations in Z/mZ recording Sep 28 inverses in Z/mZ, Z/pZ is a field, Wilson's Theorem recording Sep 30 Wilson's Theorem, Fermat's little Theorem, Check digits (no sound after 22 minutes, sorry!) recording Oct 5 ISBN-10 and ISBN-13, some homework problems recording Oct 12 Chinese Remainder Theorem recording Oct 14 polynomial equations, Hensel's Lemma recording Oct 19 comments on the exam, proof of Hensel's Lemma and example recording Oct 21 Euler's Theorem, phi function recording Oct 26 ring isomorphisms, summatory function, number and sum of divisors recording Oct 28 summatory function of phi, homework problem recording Nov 2 order, primitive roots recording Nov 4 Hensel's Lemma example, Langrange's Theorem, existence of primitive roots mod primes recording Nov 9 discrete logarithms (recording stopped after 30 minutes, sorry) recording Nov 11 symmetric encryption, Diffie-Hellman recording Nov 16 public key cryptography, signatures recording Nov 18 finite continued fractions recording Nov 23 infinite continued fractions recording Nov 30 approximation of irrationals by continued fractions, Pell's equation recording Dec 2 Pell'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).

 Homework due date remarks 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 14 fixed 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 4 added some hints on Oct 29 Homework 10 (TeX, solutions) Nov 11 Homework 11 (TeX, solutions) Nov 23 added 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.

• Midterm exam: Thursday, October 7, 14:00 - 15:30 (solutions)
• Final exam: Thursday, December 9, 9:00 - 11:00, PAR 201 (solutions)

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.