18.781 - Theory of Numbers - Fall 2007


| Schedule | (Up) Dates | Homework | PARI | Further Reading | Handouts |

General Information

Time MWF, 12-1 p.m.
Location Building 2, Room 102
Professor Ben Brubaker (brubaker@math.mit.edu)
Office: 2-267
Office Phone: 3-4079
Office Hours: W 2-4, F 10-11, and always available by appointment.
TextbookAn Introduction to the Theory of Numbers, by Niven, Zuckerman, and Montgomery (5th Ed.)
Homework -- 20%, 1 Midterm (In-Class) -- 20 %, 2 Mini-Midterms (In-Class) -- 10% Each, Final -- 40%
Number theory is becoming an all-encompassing term in modern mathematics, so where to begin? Answer: quadratic reciprocity. That is, the first few weeks, we'll work our way up to a proof of quadratic reciprocity. After that, we'll study techniques to handle other diophantine equations, discuss the primes and how to count them, and hopefully end the course with time to talk about more advanced topics, such as elliptic curves and modular forms.
Prereqs There are no prerequisite courses for 18.781, but some level of mathematical maturity will be helpful, as we'll regularly be discussing proofs in class and assigning proofs as homework. A substantial goal of the course is to strengthen these skills, so a willingness to work hard may be the only necessary prerequisite.

Announcements & Dates

  • Important Dates
    • Wednesday, September 5: First Day of Class
    • Monday, September 24: Student Holiday -- No Class
    • Friday, October 5: ADD Date
    • Monday, October 8: Columbus Day Holiday -- No Class
    • Wednesday, October 17: FIRST MIDTERM -- In Class
    • Monday, November 5: MINI MIDTERM I -- In-Class
    • Monday, November 12: Veteran's Day Holiday -- No Class
    • Friday, November 30: MINI MIDTERM II -- In-Class
    • Wednesday, November 21: DROP Date
    • Thur-Fri, November 22,23: Thanksgiving
    • Wednesday, December 12: Last Day of Class
    • December 17-21: Final Exam Week
        OUR FINAL: TBA


PARI is a computer algebra system written especially for number theory computations. We'll be using it in class to test conjectures and experiment. It's also a great way to double check your homework. It is FREE, and can be downloaded at the following main site:

  • PARI/GP Main Page (click on the "Download" link on the left margin)
  • Here are some instructions for downloading the files from the above site using UNIX:
  • Unix Installation Instructions
  • If you have a Windows-based computer, then the installation should be automatic upon downloading the appropriate file from the PARI/GP page (though I haven't tried it). For Mac users, it may be easiest to download the program Fink and Fink Commander from the web first. These programs fetch all of the appropriate files for you and then compile them on your hard drive. You can then run PARI from the terminal window (in OS X).

    Here is some additional documentation:

    You can also install PARI using your personal Athena file space, decompressing the downloaded file in /mit/username. If your personal space is full, we can even request a course locker for 18.781 with a gigabyte of space and place it there.

    Other Computing Options

  • If you have other computer-algebra software that can do the same computations, such as Mathematica or Magma, then you may use it for the course. However, PARI is the language I will use in in-class examples, so you should probably have some minimal programming experience in order to write and de-bug programs in your chosen computer-algebra system.
  • Sage is another free computer algebra system started by William Stein, now at Univ. of Washington. It includes a PARI interface (as well as many other programs). You can download it from the site above or even use an online notebook.
  • Further Reading

  • A paper by Emma Lehmer on Rational Reciprocity Laws, is a very readable account of some other attempts to generalize quadratic reciprocity and the Legendre symbol.
  • Textbooks

  • The book I almost assigned for this class was also called: "An Introduction to the Theory of Numbers," (5th Ed.) but by Hardy and Wright. I don't think the title similarities are an accident, but rather a sort of homage to the classic number theory book of all time. You'll notice by browsing the tables of contents that even many of the chapter titles are the same as those in Niven et. al. Hardy and Wright is a treasure trove of number theory facts, and you may also get additional perspective on topics we cover in class by reading Hardy as a second source. (The only reason I didn't assign it as a course text is that it is somewhat less user-friendly and contains no exercises, so it isn't quite set up to leave you to do some of the work.)
  • A Classical Introduction to Modern Number Theory by K. Ireland and M. Rosen is a terrific book for the ambitious student looking for a self-guided tour of the subject. It starts off reasonably slowly and builds to the very frontier of modern mathematics by the appendices, and all in a comprehensible way. We'll borrow passages from this book throughout the course.
  • Web Links

  • Weil's Letter on Analogy in Mathematics, appearing in the AMS Notices, 2005.
  • For that matter, the whole AMS Notices website is a good place to get the latest math news.
  • The Number Theory Web is another website for all things number theory, including many links to other sources of information.
  • Doing mathematics is a little like...

  • trying to write this poem. I find the first two paragraphs are best.
  • the tale of the horse broker in the introduction to Salinger's "Raise High the Roof Beam, Carpenters," which you can find with a little searching on the web.
  • Solutions and Handouts:

  • Solutions to the Analytic Number Theory Quiz
  • Solutions to the Reciprocity Quiz
  • Dirichlet's Theorem for the general modulus, mostly concerned with how to define a character for the general modulus.
  • An outline of Dirichlet's proof and a discussion of the evaluation of the Gauss sums. (Note, numbering of steps not the same as our in-class lecture on Monday, Nov. 26.)
  • Notes on Analytic Number Theory, covering the proof of infinitely many primes in an arithmetic progression, as presented in class on 11/14 - 11/19.
  • Notes (Part IV) covers our final discussions of algebraic integers and the problem of general reciprocity (Class dates: Nov. 7,9).
  • Notes (Part IA) were from last Friday (Oct. 26) but were left out of the previous sets.
  • Notes (Part III) which carry you up to the end of the proof of cubic reciprocity.
  • An example of how to compute with the cubic residue symbol.
  • Notes (Part I) on algebraic number theory relating to cubic reciprocity.
  • Notes (Part II) on algebraic number theory.
  • Sample questions for Midterm I. Some of these questions should resemble questions on the midterm, but clearly this is way, way too many questions for an hour.
  • List of Proofs to know for Midterm I.
  • PARI tutorial on RSA with big primes
  • In-class PARI9-12-07 worksheet as a PDF file.
  • In-class PARI9-5-07 worksheet as a PDF file.
  • Eventually, I may post further exam solutions and in-class handouts here. For partial solutions to the homework, see the homework page.