Number Theory

Number theory, "the queen of mathematics", concerns the study of the integers and their arithmetical properties. We will cover many connected ideas typically seen in a first course, including primes and factorization, the Euclidean algorithm, linear congruences, primitive roots, the theorems of Fermat, Euler and Wilson, quadratic reciprocity, public-key cryptography, sums-of-squares, continued fractions, and selected topics from algebraic and analytic number theory and arithmetic geometry such as Gaussian arithmetic, diophantine geometry, and asymptotics of arithmetic functions.

Throughout, we will focus on how to formulate mathematical ideas and conjectures about the objects we meet and learn how to write good proofs verifying our claims. While this is not a course specifically designed to teach ``proofs'', students who successfully complete the course will gain proficiency in proof-writing as well as many skills that will prove useful in any advanced mathematics course.

None required, but the following textbook is recommended:
*A Friendly Introduction to Number Theory* by J. Silverman (2012).

Another nice book is
*Number Theory and Geometry: An Introduction to Arithmetic Geometry.* by A. Lozano-Robelo (2019).

For a different style of exposition, try:
*The Higher Arithmetic* by H. Davenport (2008).

If you prefer a more ``traditional'' style of mathematics text, try:
*An Introduction to the Theory of Numbers* by I. Niven, H. Zuckerman, H. Montgomery (1991).

Instructor | Jack Petok |
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Class | MWF 11:30 - 12:35 |

X-hour | Tu 12:15 - 1:05 |

Instructor | Jack Petok |
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Office Hours | Tuesday 1:30-3:30pm; Friday 4-5pm |

jack.petok AT dartmouth.edu |

You are expected to attend class in person unless you have made alternative arrangements due to illness, medical reasons, or the need to isolate due to COVID-19. For the health and safety of our class community, please: do not attend class when you are sick, nor when you have been instructed by Student Health Services to stay home. Sick days will not count against your participation score.

In accordance with current College policy, all members of the Dartmouth community are required to wear a suitable face covering when indoors, regardless of vaccination status. This includes our classroom and other course-related locations, such as labs, studios, and office hours. If you need to take a quick drink during class, please dip your mask briefly for each sip. Eating is never permitted in the classroom. (The only exception to the mask requirement is for students with an approved disability-related accommodation; see below.) If you do not have an accommodation and refuse to comply with masking or other safety protocols, I am obligated to assure that the Covid health and safety standards are followed, and you will be asked to leave the classroom. You remain subject to course attendance policies, and dismissal from class will result in an unexcused absence. If you refuse to comply with masking or other safety protocols, and to ensure the health and safety of our community, I am obligated to report you to the Dean’s office for disciplinary action under Dartmouth’s Standards of Conduct. Additional COVID-19 protocols may emerge. Pay attention to emails from the senior administrators at the College.

There will be two timed in-person midterm exams and one in-person final during the final exam period.

If you have a question about how your exam was graded, you can ask your instructor; to have your exam regraded, please submit your question in writing to your instructor.

The only way to really figure this stuff out is to work out some exercises on your own. In a typical week, you will be assigned around 6-10 homework problems, announced on Canvas. Homework will typically be released each Wednesday and due the following Wednesday at the beginning of class. If you are experiencing an illness with severe symptoms, I will accept a high-quality scan by email. I will not accept any late homeworks, but I will drop your lowest score.

Collaboration is an important part of learning and doing mathematics. You are encouraged to discuss these problems amongst each other, but {\bf the final write-up must be your own}. Solutions to problems must be written up in a self-contained way and the write up must contain all crucial steps and not just the final answer. You are allowed to use textbooks and notes from class, and you are allowed to use other online reference and educational sources such as Wikipedia, but you are not allowed to specifically look up solutions of the homework problems as a means to avoid thinking about the problem yourself . You are also not allowed to ask for the solution by posting a particular problem on any online Q \& A site or help forum. I would advise against the use of calculators and software when doing most of the homeworks, as these will not be allowed on the exams. However, there may be a few homework problems where I will encourage the use of software/calculators. <\p>

I expect you to attend every class and to arrive on time. While there is no requirement to attend every single class, students who miss more than 5 classes without making prior arrangements with the instructor should expect to see a drop in their participation score.

It is your responsibility to keep informed of any announcements, syllabus adjustments, or policy changes made during scheduled class time. Not all announcements will be posted on Canvas, though I will try my best.

You should read all emails and Canvas announcements from the instructors and TAs. You are responsible for the policies established in any electronic communications for this class, including emails, the web page, and the Canvas page

I borrowed this next blurb from my PhD advisor: "In my experience as a student, most people do not follow all the details of a lecture in real time. When you go to a math lecture, you should expect to witness the big picture of what's going on. You should pay attention to the lecturer's advice on what is important and what isn't. A lecturer spends a long time thinking on how to deliver a presentation of an immense amount of material; they do not expect you to follow every step, but they do expect you to go home and fill in the gaps in your understanding. Not attending lecture really hurts your chances at a deep understanding of the material."

Do not fall behind. It is completely normal to not understand the lecture or not be able to solve a homework problem on your own. Attend lectures and try to follow the big picture even if you are get lost in the details. Seek help whenever you need it. There are many resources available to ensure that you are keeping up with the course. I encourage you to ask for help from your classmates and to come to my office hours whenever you are having trouble understanding the course material. I firmly believe that there is no such thing as a stupid question. Besides, asking questions in class and in office hours will boost your grade (see "participation" above).

Academic integrity is at the core of our mission as mathematicians and educators, and we take it very seriously. We also believe in working and learning together.

Collaboration on homework is permitted and encouraged, but obviously it is a violation of the honor code for someone to provide the answers for you.

On written homework, you are encouraged to work together, and you may get help from others, but you must write up the answers yourself. If you are part of a group of students that produces an answer to a problem, you cannot then copy that group answer. You must write up the answer individually, in your own words. A good practice is to discuss ideas on a blackboard, then erase the blackboard and try to reproduce the arguments later, on your own paper, and without assistance. Permitted resources for homeworks include textbooks and notes from class, and you are allowed to use other online reference and educational sources such as Wikipedia, but you are not allowed to specifically look up solutions of the homework problems as a means to avoid thinking about the problem yourself . You are also not allowed to ask for the solution by posting a particular problem on any online Q & A site or help forum.

On exams, you may not give or receive help from anyone. Exams in this course are closed book, and no notes, calculators, or other electronic devices are permitted.

Plagiarism, collusion, or other violations of the Academic Honor Principle will be referred to the Committee on Standards.

The course grade will be based upon reading and class participation, the scores on the exams, homework, and the final exam as follows:

Written homework | 25% |

Participation | 5% |

Midterm 1 | 20% |

Midterm 2 | 20% |

Final Exam | 30% |

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with your instructor before the end of the second week of the term to discuss appropriate accommodations.

Students who need academic adjustments or alternate accommodations for this course are encouraged to see their instructor privately as early in the term as possible. Students requiring disability-related academic adjustments and services must consult the Student Accessibility Services office (Carson Suite 125, 646-9900, Student.Accessibility.Services@Dartmouth.edu). Once SAS has authorized services, students must show the originally signed SAS Services and Consent Form and/or a letter on SAS letterhead to their professor. As a first step, if students have questions about whether they qualify to receive academic adjustments and services, they should contact the SAS office. All inquiries and discussions will remain confidential.