MATH567, Experimental Number Theory

Fall 2016

Instructor: Qing Xiang

Course Description.

    Number theory is a vast and fascinating field of mathematics consisting of the study of properties of the integers. As Joe Silverman said in his book "A Friendly Introduction to Number Theory", "Number theory is partly experimental and partly theoretical. The experimental part normally comes first; it leads to questions and suggests ways to answer them. The theoretical part follows; in this part one tries to devise an argument that gives a conclusive answer to the questions." In this course, we will emphasize the experimental nature of number theory, doing a lot of experiments by using SAGE.

Prerequisites

    MATH210 (Discrete Math I) and MATH245 (Introduction to Proofs), or by permission of the instructor. If you had MATH451 (Abstract Algebra), that will be great. But this is not compulsory.

Synopsis

  1. Divisibility, gcd, the Euclidean algorithm (at an advanced level),
  2. Congruences, primitive roots, the discrete log problem,
  3. Quadratic residues and the quadratic reciprocity law,
  4. The RSA public-key cryptosystem,
  5. Primality tests (in particular, the AKS theorem),
  6. Sum of squares, Pythagorean triples, and Pell's equation.

Reading List

  1. A Friendly Introduction to Number Theory, 4th edition, Joseph H. Silverman
  2. Elementary Number Theory: Primes, Congruences, and Secrets, William Stein
  3. The High Arithmetic, An Introduction to Number Theory, H. Davenport