Syllabus Math 462: Number Theory

"Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful... I know numbers are beautiful. If they aren't beautiful, nothing is." -Paul Erdos

Class Objectives

Number theory is the story of how a structure, as straight forward as the integers, can lead to incredible intellectual riches. Around every corner there is a tantalizing pattern or relationship to be discovered. It is no wonder that number theory has been the playground of some of the greatest mathematical minds of all time. In this course we will explore the basic features of the integers, and discover the beautiful realm defined by their abstraction. Along the way we will consider some of the powerful modern applications of the subject, demonstrating that even the most abstract mathematical ideas can have future applications. Through this course you will

• increase your content knowledge

• strengthen your ability to analyze and compose mathematical proofs,

• develop your mathematical maturity, independence, and Python programing skills by asking good questions, exploring examples numerically, forming conjectures, and conducting mathematical research

• practice your collaboration skills and participate in the mathematics community

Class Engagement: Students are expected to attend 4 of the math colloquium talks given throughout the semester. A schedule of talks can be found on the Math Department website, and linked to from the course WISE site. The goal of this attendance requirement is to introduce you to the diversity and vitality of current mathematics research, and to include you in the Math Department community. In addition, there will be in-class investigations, and out of class Perusall readings (including texts, podcasts, and videos) assigned to supplement class discussions. Your grade for this component of the course will be based on how meaningfully you engage in these activities and discussions, as based on your attendance and the quality of your in-class and Perusall contributions. Perusall assess how integral your contributions are to the conversation – hence engaging with the Perusall readings early and often will increase your score.

Individual Exams: There will be two individual exams (celebrations? denouements? an assessment by any other name is just as exciting). These will consist of both an in-class portion and an out of class take-home or oral exam component. More information will be given the week before the exam time.

Independent Investigation: The best way to understand new mathematical concepts and definitions is to apply them to an investigation of your own. To that end, you will be involved in your own research investigation this semester. During the first week I will give you your very own diophantine equation to study/investigate/explore. As we introduce each new definition, theorem, or technique, your job is to apply this new concept to your equation or to a related quadratic field. You will record your investigations in a research journal. For the last half of the semester, your research journal will record your independent investigation of one of the topics included in chapter 9 of the text. For information on what to include in your journal and how it will be graded, see the https://sites.google.com/willamette.edu/ntinvestigation/home

Homework & Python Worksheets: Throughout the semester I will assign homework sets and python worksheets which preview or reinforce class concepts, provide the opportunity to explore and formulate conjectures or apply the theorems and techniques discussed in class. I encourage you to do these homework sets collaboratively, though each student needs to compose and turn in their own written solutions.

Student Responsibility

I will do everything in my power to help you learn, but it is up to you to make the most of this experience by coming to class prepared and ready to engage in discussion. The following actions are all critical for your success in this class. If there are barriers to you doing any of the following, please let me know.

ASK QUESTIONS & THINK CRITICALLY. Throughout the course you should be asking yourself “Why are we doing this? Why does this method work? How is this related to other topics I’ve learned?” Ask questions in class, after class, during office hours, whenever! If you are confused or having problems with a certain section of the material talk to me about it. It is best to clear up confusions right away, and if you have a question, almost certainly other members of class do to. Be the brave one and speak up! The whole class will benefit from your questions.

DO THE ASSIGNMENTS. As you know, mathematics is not a spectator sport. At the same time, it can be a frustrating endeavor to engage in alone. Think of it as a team sport! I strongly encourage, and may even require, you to work with your fellow students on homework assignments.

SPEND TIME STUDYING. Willamette’s Credit Hour Policy holds that for every hour of class time there is an expectation of 3 hours of work outside of class. You should anticipate spending approximately 9 hours outside of class engaged in course-related activities. To master the content of this course you will need to invest time and effort. Set aside time for both homework and studying. A good approach would be to spend approximately 1-2 hours on reading the sections and making study notes, 3-6 hours doing the assigned problems and worksheets, and 1-3 hours on your research project each week. This course is completely manageable, and you can do well in it as long as you engage from the beginning. The material builds on itself making it almost impossible to follow new material if you don’t have a firm grasp on the previously covered concepts. Furthermore, your research will be most effective if you make steady progress throughout the semester.

How to Get Unstuck

Number Theory has some of the most beautiful - and complex - definitions and results you will run into in your undergraduate mathematics career. It is natural to feel a little lost or confused at times. What should you do? Discuss! Ask questions! You can start conversations on the course Discord channel, bring questions up in class, email me, arrange to meet with me, arrange to meet with other students, attend office hours, etc. We now have the capability to meet in person or virtually, so there are even more ways to talk things through with me or your classmates.

Course Policies

Inclusivity Statement: Willamette University values diversity and inclusion; we are committed to a climate of mutual respect and full participation. My goal is to create a learning environment that is effective, equitable, inclusive, and welcoming. If there are aspects of the instruction or design of this course that result in barriers to your inclusion, assessment of your work, or your achievement, please notify me as soon as possible. I will gladly honor your request to address you by your affirmed name or gender pronoun. Please advise me of this at any point in the semester so that I may make appropriate changes to my records. If I accidentally use an incorrect gender pronoun or mispronounce your name when addressing you, please let me know, in whatever manner makes you comfortable, so that I can make every effort to correct that error.

Students with disabilities are encouraged to contact the Accessible Education Services office in Smullin 155 at 503-370-6737 or Accessible-info@willamette.edu to discuss a range of options to removing barriers, including accommodations.

Missed Classes: Please make every effort to attend all classes and be an active participant in class activities and discussions. Much of your learning and understanding of new mathematical ideas will come from thinking and talking about them in class. If for some reason you are unable to attend class or turn in an assignment, please let me know as soon as possible, preferably before the missed class or assignment. I know how important athletic commitments, personal events, and religious traditions can be, and will do my best to accommodate your personal needs.

Cell Phone/Screen Policy: Unless specifically stated for class use, no phones, laptops, iPads, or other devices which take your eyes off your fellow classmates and the class discussion are allowed. Electronic devices must be turned off or silenced during class meetings. If your phone goes off, or it is clear that you are using one of these devices during class, you will be responsible for bringing treats for the entire class at the next class meeting.

Health: As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and reduce your ability to participate in daily activities. Willamette services are available and treatment does work. If you think you need help, please contact Bishop Health as soon as possible at http://willamette.edu/offices/counseling/. Crisis counseling is available 24/7 at WUTalk: 503-375-5353 and Campus Safety is available at 503-370-6911. Emergency resources are also available from the Psychiatric Crisis Center at 503-585-4949 and the National Suicide Prevention Lifeline at 1-800-273-8255.

Willamette is a community committed to fostering safe, productive learning environments, and we value ethical sexual behaviors and standards. Title IX and our school policy prohibit discrimination on the basis of sex, which regards sexual misconduct including discrimination, harassment, domestic and dating violence, sexual assault, and stalking. We understand that sexual violence can undermine students academic success, and we encourage affected students to talk to someone about their experiences and get the support they need. Please be aware that as a mandatory reporter I am required to report any instances you disclose to Willamette’s Title IX Coordinator. If you would rather share information with a confidential employee who does not have this responsibility, please contact our confidential advocate at confidential-advocate@willamette.edu. Confidential support also can 4 be found with SARAs and at the GRAC (503-851-4245); and at WUTalk - a 24-hour telephone crisis counseling support line (503-375-5353). If you are in immediate danger, please call campus safety at 503-370-6911.

Academic Integrity: Students of Willamette University are members of a community that values excellence and integrity in every aspect of life. As such, we expect all community members to live up to the highest standards of personal, ethical, and moral conduct. Students are expected not to engage in any type of academic or intellectually dishonest practice and encouraged to display honesty, trust, fairness, respect, and responsibility in all they do. Plagiarism and cheating are especially offensive to the integrity of courses in which they occur and against the College community as a whole. These acts involve intellectual dishonesty, deception, and fraud, which inhibit the honest exchange of ideas. Ignorance of what constitutes plagiarism shall not be considered a valid defense. Plagiarism and cheating may be grounds for failure in the course and/or dismissal from the College. For more information, see http://willamette.edu/cla/catalog/policies/plagiarism-cheating.php. If you are unsure of what constitutes cheating, please ask me. These are the practices I expect you to follow in each of the components of the course:

on the readings, research journal, and homework assignments: You may, and are encouraged to discuss things with fellow students, and get help from your professor, textbook, notes, or calculator. However, your submitted written work should be your own or a collaboration with your assignment group. Copy/pasting sections of another's assignment or code, reading and paraphrasing another source, or providing your assignment to be copied by others, are violations of university policy. When working on a group assignment, all members of the group should contribute to producing all components of the finished product. Writing your name on work you did not contribute to in a substantive manner is a violation of university policy.

on individual tests: Throughout the course there will be individual tests and/or quizzes. Your work on these assessments should be your own, and you may not receive help from any source other than your professor (unless directed otherwise).

Intellectual Property & Privacy: Class materials and discussions including recorded lectures are for the sole purpose of educating the students enrolled in the course. The release of such information (including but not limited to directly sharing, screen capturing, or recording content) is strictly prohibited, unless the instructor states otherwise. Doing so without the permission of the instructor will be considered an Honor Code violation and may also be a violation of other state and federal laws, such as the Copyright Act. 5

DACA/Undocumented Student Advocate: Willamette is committed to supporting our DACA/Undocumented students in a variety of ways. This year, Olivia Muñoz is the contact person for all DACA/undocumented students can provide those students with a number of external and internal resources that are available. Her contact information is email:omunoz@willamette.edu, Office: 3rd Floor UC, Phone: 503-370-6447.

SOAR Center Offerings (Food, Clothing, and School Materials): The Students Organizing for Access to Resources (SOAR) Center strives to create equitable access to food, professional clothing, commencement regalia, and scholarly resources for WU and Willamette Academy students. The SOAR Center is located in the Putnam University Center’s third floor, across from the Harrison Conference Room. The space houses the Bearcat Pantry, Clothing Share, and First-Generation Book Drive and is maintained by committed students and advisers. Please check www.willamette.edu/go/soar for current hours of operation and email soar-center@willamette.edu for any questions or concerns.

Land Acknowledgement: We are gathered on the land of the Kalapuya, who today are represented by the Confederated Tribes of the Grand Ronde and the Confederated Tribes of Siletz Indians, whose relationship with this land continues to this day. We offer gratitude for the land itself, for those who have stewarded it for generations, and for the opportunity to study, learn, work, and be in community on this land. We acknowledge that our Universitys history, like many others, is fundamentally tied to the first colonial developments in the Willamette Valley. Finally, we respectfully acknowledge and honor past, present, and future Indigenous students of Willamette.

Important Dates

8/29 (M) First day of classes

9/5 (M) Labor Day - No Classes

9/12 (M) A/D Deadline

10/10 (M) C/NC Deadline

10/14 (F) Mid-semester break - No Classes

10/28 (F) Withdraw Deadline

11/9-17 Registration for Spring

11/21-25 (M-F) Fall Break - No Classes

12/9 (F) Last Day of Classes

12/12 (M) Final Exam Time 2-5pm