Course MAT214

Numbers, Equations, and Proofs

This course gives an introduction to rigorous proofs and formal mathematical arguments in the context of elementary number theory.  It is a more algebraic alternative to MAT215, our introduction to rigorous proofs in analysis (calculus). Topics covered include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions and quadratic reciprocity. There will be a topic, chosen by the instructor, from more advanced or more applied number theory: possibilities include p-adic numbers, cryptography, and Fermat's Last Theorem. 

Equal emphasis is given to learning new mathematics and to learning how to construct and write down a correct mathematical argument by dividing the question into logical steps where each step is explained and justified carefully, giving references if necessary. For most students this will be a completely new and very challenging way of doing mathematics, very far removed from the process of memorizing algorithms and working through concrete calculations typical of high school math.

Topics

The main topics to be covered, time-permitting, are:

  • divisibility and primes
  • congruences and modular arithmetic
  • quadratic residues and quadratic reciprocity
  • binary quadratic forms
  • diophantine equations
  • prime number estimates and Dirichlet series
  • algebraic numbers

 

Description of classes

Classes are usually taught in a single section, in the Fall semester only. There are weekly problem sets that count for about 50% of the grade.  Depending on the instructor there will be quizzes and/or a midterm exam along with a final exam, usually take-home, for the remaining 50%. 

Notes

This course is usually followed by MAT217 in the spring semester and either MAT203 or MAT218.   MAT215 gives a better preparation for MAT218, which requires a very strong calculus background.  Most math majors take only one of MAT214 and MAT215 however.

Take a look at the classic short text The Higher Arithmetic by H. Davenport or Oystein Ore's Number Theory and its History to get a somewhat gentler, more recreational, introduction to the mathematics in this course. If you like reading Davenport and Ore, you will probably enjoy this course. If you find that you prefer a less algebraic topic, like calculus, consider MAT215 or MAT203 instead as your first Princeton math course.

Who Takes This Course
  • Primarily incoming students who want to major in math. Some will have already learned a little number theory before coming to Princeton and want to pursue these topics further. Others may be particularly interested in algebra, or just looking for a change from calculus.
  • Students who are choosing between math and physics as a major should perhaps consider MAT203 or MAT215 instead.
  • Computer science majors interested in cryptography might find this course useful, provided their math background is strong enough.
  • Others who don't want to major in math, but have a strong background and aptitude, along with curiousity about mathematics beyond calculus, sometimes take this course.
Placement and Prerequisites

A very strong aptitude for mathematics and real mathematical curiosity is essential. Do you want to

  • make your own conjectures and figure out for yourself whether a mathematical statement is true or false?
  • be able to construct a clear and convincing, even iron-clad, argument to justify a mathematical claim?
  • develop an appreciation for the intrinsic value and power of mathematical thinking, separate from considerations of real-world applications or utility?

Typically students have a 5 on the BC calculus exam together with a math SAT score of at least 750.

Students with a 5 on the AB exam know more than enough calculus to take this course -- this course uses little calculus.  However, prospective mathematics majors who took only AB calculus should take MAT215 instead so that they will have an adequate background in calculus to complete the departmental requirements in analysis.

Sample Material

In addition to the books by Davenport and Ore mentioned above, check out the Sample Questions to see what this course will be like. The first problem needs no special background beyond the definition of a prime number, so see if you can solve it. The others concern topics you will learn about if you take the course or if you do a little background reading.

FAQ
  1. How hard should I expect to work in this course?
    • It requires a steady time commitment. We expect that the weekly problem sets will take at least 3 hours to complete although this can vary quite a bit depending on your background and goals. It is quite difficult to judge how much time you will need to master the more abstract parts of the course.  To do well on exams, you need to spend a lot of time digesting the course material, learning the proofs and definitions well enough to adapt them to new situations and combine various standard ideas in new ways on an exam.   This course can easily require 10 or more hours of work per week, in addition to the time spent in class.
    •  For most students this course requires a very big adjustment both in effort expended and in your definition of success.  Be prepared to invest quite a lot of time early on, learning how to think about proofs and counterexamples and adapting old techniques to new situations, rather than chugging through computational problems just like the textbook and lecture examples.   This is a course for people who are more afraid of being bored than they are of being wrong.
  2. I have never had a course with rigorous proofs -- will this course be too hard for me?
    • It will probably be a challenge, but the course is designed to be accessible to students who are seeing proofs for the first time.   If after a few classes you still find the subject appealing, you should persevere even if you do find the problem sets to be (overly) challenging!  Go to your instructor's office hours and take advantage of the help available at the McGraw Study Halls, where enthusiastic math majors will help you learn to think like a mathematician. 
    • This course is graded on a generous curve to encourage interested students who want to give this kind of thinking a serious try without undue academic risk.   Consult your instructor for advice after the first few weeks if you are worried about the decision.
  3. How can I decide whether to take MAT214 or MAT215?
    MAT215 is probably a better choice if you want to be a math major because it gives you a better foundation in analysis.  All math majors have to take courses in real and complex analysis to complete the degree.  Number theory however is optional.  Students interested in number theory can learn more about this topic in more advanced algebra courses later on.
    • MAT214 is a good choice if you have excellent math skills and you would like to learn more about proof-based mathematics, and see what mathematics is like beyond calculus, before you settle down to major in a subject that does not use much calculus.  
    • MAT214 can be a good choice if you are trying to decide between computer science and mathematics as a major.  
  4. I can't fit this course into my schedule. Can I take this course for Princeton credit at another university?
    • Probably not.  If you want to be a math major you should take this course (or MAT215) here at Princeton.  This course (or MAT215)  will show you what it will mean to be a math major at Princeton.  The first rigorous proof course, followed by MAT217 and MAT218, sets the foundation for all the more advanced courses for math majors and you really need to be sure that this foundation is as solid as possible.  Credit for this course at another university will be granted only under very exceptional circumstances and we strongly prefer that you take this course here at Princeton.
  5. My question is not listed above. Where can I find an answer?
    • Try the undergraduate home page.   You will find links there to more information for future math majors and contact information if you need to talk to a person.  Representatives from the math department will be available at the academic expo during orientation and at freshman registration.