# Math 214

### 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 215, our introduction to rigorous proofs in analysis (calculus). 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.

Classes: are usually taught in a single section, in the Fall semester only. There are weekly problem sets that count for about 25% of the grade, a midterm exam that counts for a similar percentage and an final exam, usually a take-home, that counts as the remaining 50%.

Topics: will be chosen from the following rather exhaustive list: induction, well-ordering, divisibility, Euclidean algorithm, primes, unique factorization, π(x), prime number theorem, congruences, Euler's φ-function, Fermat's little theorem, Wilson's theorem, primes that can be written as the sum of squares, Chinese remainder theorem, cryptography, RSA, Hensel's lemma, primitive roots, quadratic residues, Legendre symbol, quadratic reciprocity, binary quadratic forms, class number, Polignac's formula, arithmetic functions (d, σk, ω, Ω, Δ, μ,...), Möbius inversion formula, Pythagorean triples, Diophantine equations (e.g. x4+ y4= z4), Farey sequences, Minkowski's geometry of numbers, Lagrange's four squares theorem, continued fractions, periodicity, Pell's equation. More advanced topics like Dirichlet series, elliptic curves, algebraic numbers, p-adic numbers, Fermat's last theorem may be included if time allows.

Textbook: The standard textbook is the 5th edition of An Introduction to the Theory of Numbers (Wiley) by I. Niven, H. S. Zuckerman and H. L. Montgomery.

Notes:
• This course is usually followed by 217 in the spring semester and 203 (offered only) in the fall. Students who want to strengthen their knowledge of rigorous one-variable calculus can consider following up with 215. Note that most math majors take only one of 214 and 215 however. Students with a strong background in calculus might follow 217 with 218 instead of 203.
• 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 203 or 215 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 203 or 215 instead. Exceptionally well-qualified incoming students may even be ready for 218.
• 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 760. Students with a 5 on the AB exam know more than enough calculus to take this course. A high math SAT score should indicate a good background in high school algebra as well. Your algebra will be needed more than your calculus here.

### Sample Problems

In addition to the books by Davenport and Ore mentioned above, check out these 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. To do well on exams, you need to spend a lot of time digesting the course material, learning the proofs well enough to adapt them to new situations and combine various standard ideas in new ways on an exam. All in all, this course may easily take 10 hours/week outside of class, on average. It is quite difficult to judge how much time you will need to master the more abstract parts of the course. For many students this is a big adjustment, unlike the math courses you have taken before, so be prepared to invest quite a lot of time early on, learning how to think about proofs and counterexamples and adapting old techniques to complicated new situations, rather than chugging through computational problems just like the textbook and lecture examples.
2. I can't fit this course into my schedule. Can I take this course for Princeton credit at another university?
• 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. This course (or 215) shows you what it will mean to be a math major at Princeton. It sets the foundation for all the more advanced courses for math majors, and you really need to be sure that this foundation is as strong as possible.
3. My question is not listed above. Where can I find an answer?
• Try the general math FAQ page. It should contain a section just for prospective math majors (eventually).