Math 453, Elementary Theory of Numbers

Spring 2016

MWF 2:00-2:50PM, 1 Illini Hall

Welcome to the course webpage for Math 453, Sections F13/F14. Here you will find some general info about the course. This is also the place to look for homework assignments, occasional course notes, and other things that might interest you. Some of this information will also be available through our course page on Illinois Compass.

• Our final is scheduled for Monday, May 9, 7-10PM in 1 Illini Hall (our usual classroom).
• The final will cover all the material covered in class, with no intended bias to any particular subject. This includes Chapters 1-6 and the part we did on continued fractions in Chapter 7 (finding finite and infinite continued fractions, convergents).
• I have posted some practice problems to help you study. These include more problems on the last part of the course, since you have not had as much of a chance to practice problems related to it. The solutions are here. You will also find solutions to all the homework assignments on our compass page.
• The rules for the final are: you are allowed two sides of a full regular sheet of paper of handwritten notes, no books/phones/calculators/etc except for a dictionary in book form if necessary for language reasons.
• Final Review: we will have a review for the final on Friday, May 6, 2-4PM in 141 Altgeld Hall.
• Office Hours this week are as follows: Tuesday 3-4 PM in 359 Altgeld Hall (my office); Wednesday NO OFFICE HOURS; Thursday 3-5PM in 143 Altgeld Hall; Friday 1-2PM in 359 Altgeld Hall (my office).

Course Syllabus

The textbook we'll be using is "Elementary Number Theory" by Strayer.

The prerequisite for this course is Math 347 or an equivalent proof-writing course. While 347 itself is not a strict requirement, this course does expect that the students are familiar with writing proofs.

Your grade for the course is determined as follows: 10% for homework, 25% for each of two midterms, and 40% for the final. There will be no make up exams.

Homework (along with occasional supplementary notes) will be posted here and will be due every Friday at 2PM in class, or in my office by the same day/time (if you choose the office option, you can slide it under the door). You are encouraged to work with your classmates on homework and discuss the homework problems amongst each other. However, solutions should be written up independently, and you should write down which classmates you worked with at the top of your homework. The lowest two homework scores will be dropped. Therefore the policy is that no late homework will be accepted, especially since we will sometimes discuss the solutions to the homework problems in class.

The midterms will be in class on Friday, February 26th and Friday, April 8th. More details about these exams will be announced in class and posted here closer to the dates.

• 1/20: Divisibility and infinitude of primes. Reading: beginning of 1.1, 1.2.
• 1/22: A few more cool facts about primes, division algorithm, Euclidean algorithm. Reading: 1.2 and second half of 1.1, 1.3.
• 1/25: Euclidean algorithm, revisiting gcd definition. Reading: Section 1.3.
• 1/27 Revisiting primes, fundamental theorem of arithmetic. Reading: Sections 1.4, 1.5.
• 1/29: Least common multiple, primes in arithmetic progressions. Reading: Section 1.5.
• 2/3: Finishing off primes in arithmetic progressions, congruences, linear congruences. Reading: Sections 1.5, 2.1.
• 2/5: Congruences, discussing Fermat's Little Theorem. Reading: Section 2.1.
• 2/8: Linear Congruences: Section 2.2.
• 2/10: Linear Congruences, starting Chinese Remainder Theorem. Reading: Section 2.2, 2.3.
• 2/12: Chinese Remainder Theorem. Reading: Section 2.3.
• 2/15: Wilson's Theorem, Fermat's Little Theorem. Reading: Sections 2.4, 2.5.
• 2/17: Fermat's Little Theorem continued, Euler's Theorem. Reading: Sections 2.5, 2.6
• 2/19: Arithmetic functions: nu, phi, sigma, and mu. Reading: Section 3.1.
• 2/22: Arithmetic functions, multiplicativity. Reading: Sections 3.2-3.4.
• 2/24: Review for Midterm 1, covering all material through Chinese Remainder Theorem.
• 2/26: Midterm 1.
• 2/29: Multiplicativity of phi, some results on multiplicativity. Reading: Section 3.2.
• 3/2: going over Midterm 1, multiplicativity of nu and sigma. Reading: Sections 3.2-3.4.
• 3/4: Formulas for phi, nu, and sigma. Perfect numbers. Reading: Sections 3.2-3.5.
• 3/7: Perfect numbers, beginning Moebius inversion. Reading: Sections 3.5-3.6.
• 3/9: Moebius Inversion. Reading: Section 3.6 in the book.
• 3/11: Moebius Inversion continued, starting quadratic residues. Reading: Sections 3.6 and 4.1 in the book.
• 3/14: Quadratic residues, Legendre symbol, Euler's criterion. Reading: Section 4.2 in the book.
• 3/16: Euler's criterion, Gauss's Lemma. Reading: Section 4.2 in the book.
• 3/18: Proof and consequences of Gauss's lemma, stating quadratic reciprocity. Reading: Sections 4.2 and 4.3 in the book.
• 3/28: Quadratic reciprocity, Eisenstein's lemma. Reading: Section 4.3 in the book.
• 3/30: Proving quadratic reciprocity. Reading: Section 4.3 in the book.
• 4/1: Reviewing proof of quadratic reciprocity, starting on primitive roots. Reading: Sections 4.3 and 5.1 in the book.
• 4/4: Primitive roots, order of an element mod m. Reading: Sections 5.1 and 5.2 in the book.
• 4/6: Review for midterm 2.
• 4/8: Midterm 2.
• 4/11: Primitive roots continued: number of primitive roots mod m. Reading: Sections 5.1 and 5.2 in the book.
• 4/13: Existence of primitive roots modulo prime numbers. Reading: Sections 5.2 and 5.3 in the book.
• 4/15: Primitive root theorem (NOTE: we did not go through the full proof of the PRT, but you are responsible for knowing the statement). Reading: Section 5.3 in the book.
• 4/18:Apollonian packings, Fermat's last theorem. This is not in the book. It might appear on a True/False in the final, but no more than that.
• 4/20: Pythagorean triples. Reading: Section 6.3 in the book.
• 4/22: Sums of two squares. Reading: Section 6.5 in the book.
• 4/25: Sums of two squares continued. Reading: Section 6.5 in the book.
• 4/27: Sums of three and four aquares, statements only (no proofs) and beginning finite continued fractions. Reading: Sections 6.5 and 7.2 in the book.
• 4/29: Finite continued fractions: convergents. Reading: Section 7.3 in the book.
• 5/2: Infinite continued fractions: existence and how to find. What it means to have an eventually periodic continued fraction expansion. Reading: Section 7.4, just a bit of 7.5 (on eventually periodic continued fractions).
• 5/4: More examples of finding infinite continued fractions; how well do convergents approximate numbers? Reading: Section 7.4.
• 5/6: Review for final exam: 2-4PM in 141 Altgeld Hall.
• 5/9: Final exam, 7-10PM in 1 Illini Hall.

• Homework 1, due 1/29/2016
• Homework 2, due 2/5/2016
• Homework 3, due 2/12/2016
• Homework 4, due 2/19/2016
• There is no homework due on 2/26/2016, since our first midterm is that day. Here are some problems to help you prepare. The solutions to these problems are here.
• Homework 5, due 3/4/2015
• Homework 6, due 3/11/2016
• Homework 7, due 3/18/2016
• Homework 8, due 4/1/2016
• There is no homework due on 4/8, since our second midterm is that day, but here are some problems to help you practice. Here are the solutions.
• Homework 9, due 4/15/2016
• Homework 10, due 4/22/2016
• Homework 11, due 5/4/2016. NOTE THE UNUSUAL DUE DATE! It's the last day of class.
• Here are some problems to help you study for the final. Here are the solutions.