# Math 406 Syllabus, Fall 2018

The following syllabus is tentative and updated only until the current date. It was last updated on 29 Nov. 2018.
 Month Date Topic Sections August 27 Introduction: course overview / Divisibility 1 29 The division algorithm / The gcd / the Euclidean algorithm 2.5-2.7 31 The extended Euclidean algorithm / Other bases 2.7-2.8 September 5 Other bases / The extended Euclidean algorithm / Fermat & Mersenne Primes 2.7-2.9 7 Fermat & Mersenne Primes / Review / Theorem 3.1 3.1-3.2 10 Theorem 3.1 3.1-3.2 12 The exact change problem / postage stamp problem 3.2 14 Fundamental Theorem of Arithmetic 4.1-4.2 17 Rational Root Theorem (with proof) / Pythagorean triples (no proof) 5.1-5.4 21 Riemann zeta function 5.7 24 Divergence of the sum of reciprocals of primes 5.7.1 26 Congruences - definitions and basic properties 6.1 28 Quizterm 1 October 1 Modular exponentiation / divisibility tests / Linear congruences 6.2-6.4 3 Linear congruences 6.4 5 The Chinese remainder theorem 6.5 8 Simple ciphers 7.2-7.4 10 Stream and block ciphers 7.5-7.6 12 Review Ch. 2-6 15 Quizterm 2 17 Modular polynomials information sharing / Fermat's little theorem 7.7, 8.1 19 Euler's theorem 8.2 22 Euler's theorem / Euler's phi function 8.2 24 Euler's phi function / RSA 8.4, 9.1 26 RSA / digital signatures 9.1-9.2 29 Order / application to Fermat numbers / Primitive roots 11.1-11.2 31 Primitive roots / Decimal criterion for rationality 11.2-11.3 November 2 Quizterm 3 5 The discrete log problem / Diffie-Helman / ElGamal 11.5, 12.1, 12.4 7 Squares mod p 13.1 9 Squares mod p / Statement of Quadratic Reciprocity / Primality testing and Carmichael numbers 13.1, 14.2 12 Statement of Minkowski's theorem / volume computations 15.1 14 Fermat's theorem on sums of squares 15.2 16 Conclusion of the proof of Minkowski's theorem 15.1 19 Quizterm 4 26 Elliptic curves mod p 28 Elliptic curve addition law 30 Elliptic curve addition law, discrete log problem, Diffie-Helman, ElGamal December 3 Quizterm 5 5 Associativity of the group law 7 Review - part 1 10 Review - part 2 17 Final exam