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:
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.