## MATH 347, Sections E1/G1 - Fall 2016: Temporary Website

### Announcements

• [11/17/2016] HW 9 available for pickup Friday, 11/18, 3 pm, 147 Altgeld: The graded homework, with solutions, will be available for pickup on Friday in the regular classroom for the 3 pm section. Homework assignments that have not been picked up will be returned by Prof. Heller after the break. Have a wonderful Thanksgiving Break!
• [11/14/2016] HW 9 is due Wednesday, 11/16. Here are some tips:
• Congruence problems (#1, #2, #3): All of these problems should be done using "congruence magic", not by other methods. Use the properties of congruences as in the examples from class, HW 8, and the following handouts: Solutions to Worksheet I, Solutions to Worksheet II, Mersenne and Fermat Type Numbers .
• None of the problems requires induction. With congruences, no induction is needed.
• None of the problems requires using the definition of congruences. Just use the properties.
• Proof practice with relations (#4). The proofs are all quite short and not difficult, but pay attention to the write-up. Make sure to include quantifiers where necessary. For example, in the definition of divisibility: "a|b" means "there exists m in Z such that b=ma".
• [11/11/2016] Worksheets and handouts from Summer 2014 class. This page contains all handouts and worksheets from my Summer 2014 Math 347 class, neatly organized. You are welcome to check it out and grab any materials you find useful.
• [11/8/2016] HW due Wednesday, 11/9. I'd be happy to answer any questions by email or in person (email ajh@illinois.edu). Here are some tips:
• Proof practice problems (#1 and #2): Model your proofs after those given in class and on the Number Theory I Worksheet. See the Solutions to this worksheet. In particular, make sure your proofs use the "official" definitions of divisibility and congruences given in class and on the worksheets (but only the definition, not the properties), and contain all necessary steps, in the correct logical order (i.e., the proof should start out with the assumptions stated in the "if" part, and end with the conclusion contained in the "then" part, not the other way around).
• "Congruence magic" problems (#3): Use congruences and their properties for these problems, as in the examples worked out in Monday's class. (You can use any of the properties listed on the worksheet, as well as Fermat's Little Theorem.) Finding last digits is equivalent to finding the remainder modulo 10 (or modulo the given base).

I will be covering Sections E1 and G1 of this course for Professor Heller from early November through Thanksgiving Break. This page, http://www.math.illinois.edu/~ajh/347/, will serve as a temporary course website during this period. You will find there basic information, announcements, homework assignments, and course handouts. I plan to keep this page up until the end of the semester.

### Basic Information

• Time/Room: Section E1, MWF 1:00 pm - 1:50 pm, 203 Transportation Building; Section G1, MWF 3:00 pm - 3:50 pm, 147 Altgeld Hall.
Class cancellation: To compensate for the evening midterm on Oct. 26, there will be no class on Friday, Nov. 18 (i.e., the Friday before Thanksgiving Break).
• Instructor (Nov. 2 - 16): A.J. Hildebrand. Office 13 Illini Hall, email ajh@illinois.edu.
• Office Hours: I will be available daily after each class period in the regular classroom, i.e., MWF 1:50 - 2:50 pm in 203 TB and 3:50 - 4:50 pm in 147 Altgeld (or one of the adjacent rooms if the classroom is used for another class). If you have classes during these periods, I'd be happy to meet with you at other times; email me (ajh@illinois.edu) to set up an appointment.
• Homework and grading: The homework and grading schedule will remain as announced on Professor Heller's website: Homework will be due Wednesdays in class, and returned in Friday's class. There will be two homework assignments during the period I am substituting: HW 8, due 11/9, and HW 9, due 11/16. The assignments will be announced in class and posted on this page.
• Syllabus: We will cover parts of Chapters 6 and 7 of the text; see below for more details. These chapters focus on Number Theory, and they include some the most interesting and most fun parts of the entire course!
• Official Course Webpage. Professor Heller's course website, with information on course policies, grading, and exams.
• Summer 2014 Math 347 Course Webpage. This page, from a previous Math 347 course I taught, offers a variety of handouts, worksheets, sample exams, etc. covering the entire Math 347 syllabus. You are welcome to check it out and grab any materials you find useful.

### Syllabus

We will begin the final part of the Math 347 syllabus, "Additional Topics". The goal of this part is to illustrate and practice the fundamental concepts, ideas, and proof techniques you've learned in Chapters 1-4 and 13-14 through selected topics from other chapters of the text. Our focus here will be on topics from number theory (Chapters 6 and 7 in the text), an area that, on the one hand, is an excellent proving ground for constructing and writing proofs, but on the other hand is also rich in interesting, fun, and famous problems. You will encounter famous proofs such as Euclid's proof on the infinitude of prime, famous theorems such as the Fundamental Theorem of Arithmetic, and famous numbers such as Mersenne and Fermat numbers, and you will learn what makes these numbers so special. You will also learn the story behind the famous "postmark prime" (google!), which was discovered right here at the UIUC CS Department.

Here is a tentative list of topics I plan to cover during the next two weeks:

• Divisibility
• Primes and composite numbers
• Euclid's Theorem on infinitude of primes
• Fundamental Theorem of Arithmetic
• Congruences and their applications
• Relations and equivalence relations
I will provide handouts with the basic definitions and theorems for these topics, as well as worksheets with practice problems.

### Class Diary, Handouts, and Worksheets

• Wednesday, 11/16:
• Monday, 11/14:
• Class: Relations, continued. Equivalence relations, equivalence classes.
• Friday, 11/11:
• Wednesday, 11/9:
• Class: More applications of congruences: Primality/compositeness of Mersenne and Fermat type numbers. Divisibility tests.
• Cool link: Online Encyclopedia of Sequences (OEIS). This may be the most amazing mathematical website. It is the Google of mathematical sequences. Try it out: For exmple, type 3,5,17,257,65537 into the search box.
• Monday, 11/7:
• Friday, 11/4:
• Class: More on the Fundamental Theorem of Arithmetic. Why 1 (and -2, -3, etc) is not a prime. Euclid's proof of the infinitude of primes. Congruences.
• Handout:Number Theory II: Congruences"
• Cool link: WolframAlpha. The Google of mathematical knowledge, hosted right here in Champaign. Sample queries: "is 1 a prime?", "factor 123456", "divisors of 347347".
• Wednesday, 11/2: