MAT 5400 (Elementary Theory of Numbers)
Fall 2020, Section 001, CRN 13433
Professor Drucker

Credits: 3  Prerequisites: MAT 2030 and 2250, or consent of instructor.
Class structure: Due to public health concerns caused by the Covid-19 pandemic, this class will be taught in an online format during Fall 2020.
Important dates this semester: Classes begin Tue., Sep. 1. Late registration and 1st week late adds, Tue., Sep. 1 – Tue., Sep. 8. Late registration and 2nd week late adds (dept permission required), Wed., Sep. 9 – Tue., Sep. 15. There will be no classes on Mon., Sep. 7 (Labor Day). The last day for tuition cancellation is Tue., Sep. 15. The last day to drop classes and not have them appear on your record is Wed., Sep. 15; after that, instructor approval is needed to drop classes. The last day to withdraw is Sun., Nov. 15. There will be no classes on Tue., Nov. 3 (federal election holiday) or from Wed., Nov. 25 through Sun., Nov. 29 (Thanksgiving break). Classes end (not counting the final exam period) on Mon., Dec. 14. Tue., Dec. 15 will be a Study Day. Final exams run from Wed., Dec. 16 through Tue., Dec. 22.
Religious holidays: Because of the variety of religious affiliations of the University student body and staff, the WSU academic calendar makes no provisions for religious holidays. However it is university policy to respect the faith and religious obligations of the individual. If anything in this course conflicts with your religious observances, please notify me well in advance so that we can work out a mutually agreeable arrangement.

Instructor: Daniel Drucker, Professor of Mathematics  Office: 1193 FAB, but I’ll be working from home this semester because I’m in a high risk age group for Covid-19.
Home page: https://drucker.wayne.edu
E-mail (best way to reach me): ddrucker [at] wayne [dot] edu   
Phone: If you are in a crisis situation, call the Math Dept at (313) 577-2479, ask the staff member for help, and ask him/her to contact me. You can also send an email to math [at] wayne [dot] edu.
Online group discussion hours: Schedule to be determined. These are for review and reinforcement, examples, homework Q&A, feedback, etc. I expect you to take part with your camera on so that I will know who is speaking and not be limited to hearing a disembodied voice.
Help by email: I will try to answer questions that you send me by e-mail, though it is sometimes hard to discuss calculations that way because of the difficulty of typing mathematics in emails.
Private individual online office hours: By appointment.

Text: John J. Watkins, Number Theory: A Historical Approach, (First Edition), Princeton University Press, 2014, hardcover, ISBN 978-0-691-15940-9, publisher’s list price $82.50, Barnes & Noble Bookstore price $82.50 new, $61.90 used; eBook, ISBN 978-1-400-84874-4, Barnes & Noble price $78.50. You can rent a new book from Barnes & Noble for $73.00, a used book for $34.65, or an eBook for $75.00, but you have to return it by 12/22/2020. I recommend that you not rent, since by paying a little more you get to keep the book for future reference. (Why would you rent an eBook for $75 when you can buy it for $78.50?) The last time I taught this course, an eBook rental was only $41. I don’t know why it’s so much more expensive now.
NOTE: The Barnes & Noble Bookstore will price match Amazon, bn.com, and local competitors. For details, click on We Price Match.
     If you order online, be aware that the shipping time starts from the time the item is sent out, which may be many days after you place your order.

Videos: We’ll be viewing the Introduction to Number Theory videos by Professor Edward B. Burger of Williams College. This set of 24 half hour videos is non-technical and accessible. It overlaps the material in our text, but also discusses some interesting applications and indicates where the material in our course leads.

About number theory: In this branch of mathematics, the objects (integers) are familiar, but the material is sometimes surprisingly abstract. At other times, it is purely computational. Initially, you may find that you feel quite comfortable when you can compute with specific numbers, but uncomfortable when you must reach conclusions or write proofs about integers in general. Likewise, if you are not used to reading abstract mathematics, you may have some trouble reading parts of the text on your own, especially the proofs. This is normal and not necessarily a cause for alarm. I will do my best to bring you “up to speed” with supplementary notes and/or videos, and in discussion sessions and/or office hours. All I ask is that you work hard to overcome any obstacles you encounter. Just as in learning to ride a bicycle or to play a musical instrument, you will probably find that you don’t improve steadily, but in jumps separated by frustrating periods of slow progress. Watkins’ text tries hard to help you understand how the developers of the subject approached it.

Materials: You will need a desktop or laptop computer with reliable internet access and audio (microphone, and speakers or headsets) in order to watch video lectures and participate in online discussions. You will also need access to a webcam and scanner in order to upload written tests and problem solutions (in pdf format, please). Needed software includes a web browser such as Chrome, Firefox, or Safari; Canvas, Zoom, and MS Teams (free to students); and perhaps supporting applications like Java, Flash, QuickTime, and Adobe Reader (or Preview on a Mac). We will use the Zoom (and perhaps the MS Teams) videoconferencing applications through Canvas for office hours and discussion groups. Most smartphones can run Zoom and serve as scanners. (Download a free app like AdobeScan).

Calculators and computers: Use of calculators and computers on assignments is encouraged, provided you use them for exploration and numerical checks, rather than as a substitute for understanding. The free online programs Sage and WolframAlpha can be useful for checking your calculations and for numerical explorations. There are also many free Java applets online that you can use for the same purpose. Calculators and computers will not normally be allowed on quizzes, so you will need to practice doing calculations by hand. In fact, one goal of the course is for you to learn to avoid using methods that force you to do calculations with unnecessarily large numbers. (Problems requiring calculation with large numbers won&rwquo;t appear on quizzes.)

Some course learning outcomes:

1. Students will learn to prove formulas and inequalities depending on an arbitrary positive integer n by use of mathematical induction.
   
2. Students will learn to use the Euclidean algorithm to solve verbal problems equivalent to linear Diophantine equations with side conditions.
   
3. Students will learn to use basic theorems in combination with modular arithmetic to calculate with large numbers while actually working with small numbers, and apply the calculations to solve congruences. They will also apply modular arithmetic to divisibility and primality tests.
   
4. Students will learn to apply the law of quadratic reciprocity and Legendre symbols to solve quadratic congruences.

Expectations: In addition to reading the text and my detailed notes that supplement the text and video lectures, you will be required to

1. submit a short introductory video (five minutes or less) including your name, interests, goals, pets, equipment to be used for the course (desktop? laptop? tablet? smart phone?), concerns about the course, and any problems you’re dealing with, but also a little about your background, your family, your pets, etc.—whatever you feel comfortable talking about
2. watch recorded video lectures each week
3. submit solutions to assigned problems on schedule
4. submit a weekly journal in the form of a short video (no more than five minutes long) or a written report, describing what you have learned that week, how well you’re keeping up, any problems (mathematical, technological, and/or personal) you have encountered, what strategies you will use to try to overcome those problems, and any suggestions you might have for making the course work better for you
5. participate in one or more discussion sessions each week with your camera and microphone turned on
6. take a number of timed quizzes, perhaps one for each chapter of the text that we cover (though a chapter heavy in content might necessitate more than one quiz)

Assigned problems: The assigned problems will be posted on the MAT 5400 Assigned Problems web page. Some will be submitted and graded as we complete each segment of the course. Assigned problems will be graded both on the correctness of your solutions and on their clarity and completeness. In particular, an answer without an explanation will receive no credit. You will see many examples of written solutions in the text. Most assigned problems will be from the text, but I will add some problems and/or projects of my own. Further advice and information about assigned problems can be found on the Assigned Problems web page.

Tests: There will be a number of quizzes, each covering at most one or two chapters of the text. There will not be any final exam. Write your solutions on lined or unlined pages with straight edges and send them to me as scans in pdf format. Leave wide margins so that there is no writing near the edges that a scan can miss. Leave room in the margins and below each solution for me to write comments. The problems and problem parts should be clearly labeled and the pages should be labeled with your name and numbered in the correct order.
Final exam slot: To be determined. We may use this slot (or one near it in time) for a final discussion and/or a last quiz.

Course grade: Taken together, the quizzes will determine 40% of your grade. The assigned problems will account for 45% of the grade. The remaining 15% will be based on weekly journals (10%) and active participation in discussion sessions (5%).
Letter grades (as intervals of percentages): A [88, 100], A– [85, 88), B+ [82, 85), B [73, 82), B– [70, 73), C+ [67, 70), C [58, 67), C– [55, 58), D+ [53, 55), D [47, 53), D– [45, 47), E [0, 45). Qualitatively, being able to do the basic calculations is worth a C. Beyond that, a solid understanding of the main abstract concepts brings you up to the B level. For an A, you should have a firm grasp of both the concrete and abstract aspects of the course.

Dropping and withdrawing: Students can drop this class through the end of the second week and receive 100% tuition cancellation. Dropped classes will not appear on your transcript. After two weeks, students are no longer allowed to drop but must withdraw with no tuition cancellation. The transcript will show a notation of WP (passing), WF (failing), or WN (no graded work) at the time of withdrawal. I am required to give a course grade of F to any student who stops attending class after the second week without officially filing a withdrawal request via the Student tab in Pipeline. As a courtesy, I request that you notify me in advance if you decide to drop or withdraw.
     Any student intending to withdraw from a course is required to complete a SMART check. Click Dropping and Withdrawing and SMART Check for more information.

Academic dishonesty: Academic misconduct is any activity that tends to compromise the academic integrity of the institution or undermine the education process. Examples of academic misconduct include:

• Plagiarism: To take and use another’s words or ideas as your own without appropriate referencing or citation.
• Cheating: Intentionally using or attempting to use or intentionally providing unauthorized materials, information or assistance in any academic exercise. This includes copying from another student’s test paper, allowing another student to copy from your test, using unauthorized material during an exam and submitting a term paper for a current class that has been submitted in a past class without appropriate permission.
• Fabrication: Intentional or unauthorized falsification or invention of any information or citation, such as knowingly attributing citations to the wrong source or listing a fake reference in a paper or bibliography.
• Other: Selling, buying or stealing all of part of a test or term paper, unauthorized use of resources, enlisting in the assistance of a substitute when taking exams, destroying anothers work, threatening or exploiting students or instructors, or any other violation of course rules as contained in the course syllabus or other written information.

Student services: The Mathematics Resources Center (MRC) is a free tutoring service provided by the Department of Mathematics for all students currently enrolled in mathematics classes at WSU. The MRC will offer online tutoring services in Canvas for the Fall 2020 term from Wed., Sep. 2 through Thu., Dec. 17. Its tentative hours of operation are M–Th 10:00 a.m.–5:00 p.m. and F 10:00 a.m.–1:00 p.m. For more details, see the MRC web page.
    The Academic Success Center (1600 Undergraduate Library) assists students in strengthening study skills. Visit www.success.wayne.edu for schedules and information on study skills workshops, tutoring, and supplemental instruction.

Technology Support: For assistance with Canvas, see C&IT’s Student Intro to Canvas Course and Comprehensive Student Guide. For assistance with technology issues, contact the C&IT Helpdesk M–F from 7:30 a.m. to 8 p.m. at (313) 577-4357 or email helpdesk@wayne.edu. Students who lack adequate hardware or reliable internet access should email the Dean of Students at doso@wayne.edu or call (313) 577-1010 for assistance. Students on campus also have access to equipment through University Libraries. For more information, see Equipment Checkout Policy and Procedures.

Support for Online Learning:

• The Effective Online Learner is a self-paced Canvas course to bolster the skills needed for remote and online learning.
• Learn Anywhere is a webpage that provides resources and tips for taking online courses.
• Academic Success Center offers resources for improving study skills, building study groups, and similar activities.
• Study Skills Academy offers free study skills counseling and other services.

Students with disabilities: If you have a documented disability that requires accommodations, you will need to register with Student Disability Services (SDS). The SDS office is located at 1600 David Adamany Undergraduate Library in the Student Academic Success Services department. Their telephone number is (313) 577-1851, or (313) 202-4216 (videophone for deaf/hearing impaired students). Once you have your accommodations in place, I will be glad to meet with you privately online to discuss your special needs. SDS’s mission is to assist the university in creating an accessible community where students with disabilities have an equal opportunity to fully participate in their educational experience.

CAPS: It is quite common for college students to experience mental health challenges such as stress, anxiety, and depression, that interfere with academic performance and negatively impact daily life. Help is available for any currently enrolled WSU student who is struggling with a mental health difficulty, at WSU Counseling and Psychological Services (CAPS), 313-577-3398. Other options, for students and nonstudents, include the Counseling and Testing Center and the Mental Health and Wellness Clinic (MHaWC) in the WSU College of Education. Remember that getting help, before stress reaches a crisis point, is a smart and courageous thing to do—for yourself, and for those you care about. Also, know that the WSU Police Department ( (313) 577-2222 ) has personnel trained to respond sensitively to mental health emergencies at all hours.

SOME ADVICE:

Check your WSU e-mail.

I expect you to check e-mail regularly, so that I can send you written messages. Click here for information on WSU e-mail. If you do not use the WSU e-mail address assigned to you, then set up your external e-mail account to fetch copies of your incoming WSU e-mail or set up your WSU account so that it forwards e-mail to your preferred address. (See this page for advice and links to instructions.)
    IMPORTANT: When you write to me, always include “MAT 5400” in the subject line of your e-mail message, and always sign the message with your full name. I don’t want to have to figure out who you are from your e-mail address.

Read the material in the text and in my notes—carefully.

I won’t always recommend doing things the way they’re done in the text and I’ll add material that’s not in the book. You’re responsible for all the material in the sections we discuss (except material I specifically exclude).
    IMPORTANT NOTE: When reviewing for tests, give highest priority to things done only (or differently) in my notes. After that, study the text and the assigned exercises.

Keep up with the assignments.

Problems (and projects, if any) will be assigned well in advance. Do them as we get to them. Don’t fall behind! It’s tough to catch up in a math class. Solving lots of problems is the only way to learn mathematics. The assigned exercises should be regarded as the minimum you should do. Whenever possible, solve extra problems. Suggestion: Keep your solved exercises in a binder that allows you to insert and remove pages, and organize it by chapter number, section number, and exercise number. Be sure to put your name and the course number on each page.

Ask questions!

It’s your responsibility to ask about anything you don’t understand. Write down the things that bother you while you’re reading the text and notes, or working on homework, so you’ll be ready with a list of questions when you come to discussion sessions and/or office hours. You can also send me questions by email, though it can be difficult to write mathematics in emails.

Exchange contact information with some of your classmates.

It’s good to be able to discuss course material with others in the class. It helps to keep you involved and motivated. Learning all by yourself or working steadily in the midst of family distractions can be difficult.

Think about the material and explore with your calculator and/or computer software (like Sage) and/or Java applets. Discuss it with classmates.

One of your goals is to learn how to speak and write mathematics. Talking about mathematics with others and making sure they understand you is excellent practice. Another goal is to try to enjoy the process of thinking about mathematics and increasing your understanding of it. You’ll find that it’s much more satisfying to figure something out on your own than to memorize facts that someone else states for you.

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