Math 3240 - Spring 2020
Introduction to Number Theory

Syllabus: We plan to cover the following topics.

• the Euclidean algorithm
• congruences in Z
• primes (theorems and conjectures)
• Fermat's and Euler's theorems
• the Gaussian integers Z[i]
• number theory of polynomials
• the Chinese remainder theorem
• quadratic reciprocity

Prerequisites: Math 2710. You are expected to know something about writing proofs, although the course itself will provide a lot of further practice. If you did not develop in Math 2710 some skill with expressing mathematical ideas well and writing proofs, then you need to focus some serious efforts in that direction early on.

Course grade:  This will be based on the following weighting: homework (20%), quizzes (10%), first and second midterms (15% each), and final exam (40%). The lowest homework and quiz grade will be dropped. This includes a grade of 0 for a homework or quiz that is not submitted. Your lowest midterm exam score will be replaced with the average of the original score on that exam and your final exam score if it is higher (i.e., if it benefits you).

Homework: Homework assignments will be posted on the bottom of this web page, and are due at the time and place indicated on the assignment. No late homeworks will be accepted. Read the homework guidelines here and pay close attention to the rules about submissions. In particular, all homework must be prepared in LaTeX. Students will be provided with a LaTeX template file for each problem set. Information about LaTeX preparation is in the last homework bullet point below.

• An integral part of each homework is the assigned reading from the handouts and the re-reading of your lecture notes. Focus on both explanations and examples.
• Homework will be done in student groups. The procedure will be discussed during class in the first week.
• It is a mistake to skip homework, because no skills (in mathematics, foreign language, athletics, and so on) can be learned by passive involvement, but only by regular practice. Moreover, many skills are learned over time, so do not expect to understand everything perfectly right away. You should find your understanding of basic topics improving gradually from one week to the next.
• Proofs on homeworks should not be simply a string of logical and mathematical symbols, but include complete sentences in English. The role of English is to explain the strategy of your proof and the details as well. There will not be partial credit based on having misunderstood a question.
• LaTeX preparation: If you want to download LaTeX for your laptop, a good LaTeX package for PCs (using Windows) is MiKTeX version 2.9, which can be downloaded here; download and run `miktex basic installer' (it takes a while to install). In the start menu of MiKTeX 2.9 you want to look for TeXworks and use that as your TeX interface. (An introductory TeXworks website is here, explaining a bit about it.) When typsetting a document with MiKTeX 2.9 for the first time, be sure to set pdftex to pdflatex. As a front-end for MiKTeX you can use TeXnicCenter.

  To install LaTeX for a Mac use MacTeX and as a front end use Texmaker.

  Students in a homework group of more than one person can use the website Overleaf for collaborative LaTeX writing.

  Some resources for learning LaTeX are a short document by the instructor here and a longer document prepared by former UConn math major John Pawlina here. Both documents have hyperlinks in them, so if you don't see them try another browser.

  You can draw a math symbol and see its TeX code ("deTeXify it") here, and you can ask questions about LaTeX at tex.stackexchange.

Quizzes and Exams:  In weeks when homework is not due, there will be a quiz at the start of one class, usually on Thursdays. The purpose of quizzes is to provide you some feedback about how well you are following the basic ideas of the course. During the semester there will be two midterms (dates are at the top of this page).  

• The quizzes will be short.
• You are not allowed to use any aids during the quizzes or exams.
• You might be asked to bring UConn photo ID to the quizzes or exams.
• There are no makeup quizzes or exams. If you miss a quiz or exam without a properly documented reason, your grade on it is 0.
• If you need accommodations based on a documented disability, you need to speak with both the Center for Student Disabilities and the course instructor within the first two weeks of the semester.

Attendance: Since you will be working in groups, your workmates can get frustrated if you regularly skip class and then cannot meaningfully contribute to the homework. The best way to begin to learn the material is to come to class without exception, see examples and techniques discussed in real time, and ask lots of questions. The way you should think about the material will develop from the way it is presented in class.

Course conduct: To respect everyone's right to a productive learning environment, please refrain from disruptive activities during class. This includes using smartphones. Please turn off all other electronic gadgets before entering the classroom (unless you take notes electronically). On a positive note, do feel free to ask questions!

Academic integrity: Students are expected to avoid academic misconduct. Your integrity is not worth losing (and the course not worth failing) by falsely presenting yourself in any aspect of this course. For further information on academic integrity, see the Student Code.