• Lecturers: Alexander Gorodnik
• Class Time: Mon. 5-6pm at Chemistry Building: West Block Lecture Theatre 4; Wed. 9-10am at Maths Building: SM1; Fr. 2-3pm at Maths Building: SM1
• Office hours: Monday 4-5pm and by appointment

• Level: H/6
• Credit point value: 20cp
• Year: 16/17
• Prerequisites: MATH11511 "Number Theory and Group Theory" and MATH11006 "Analysis 1"

• Course Information Handout

Course Description: Number theory is a thriving and active area of research whose origins are amongst the oldest in mathematics; some questions asked over two thousand years ago have not been fully answered yet. Despite this ancient heritage, it has surprisingly contemporary applications, underpinning the internet data security that lies at the heart of the Digital Age. Although at the core of number theory one finds the basic properties of the integers and rational numbers, the subject has developed coherently in many directions as it has been influenced by (and indeed as it in turn influences) partner disciplines. Almost every conceivable mathematical discipline has played a role in this development, and indeed this web of interactions encompasses Algebra and Algebraic Geometry, Analysis, Combinatorics, Probability, Logic, Computer Science, Mathematical Physics, and beyond.

At the end of the unit you will acquire a command of the basic tools of number theory as applicable to the investigation of congruences, arithmetic functions, Diophantine equations and beyond. In addition, you will become familiar with the underlying themes and current state of knowledge of several branches of Number Theory and its interaction with partner disciplines.

Topics covered will include:

• Revision of the basic properties of the integers including the Euclidean algorithm.
• Number-theoretic functions, especially the Mobius and Euler functions. Averages and maximum values.
• Congruences, including the theorems of Fermat, Euler, and Lagrange, and computational applications. The RSA cryptosystem.
• Primitive roots and the structure of the residues modulo m.
• Polynomial congruences to prime powers. Hensel's lemma and the p-adic numbers.
• The quadratic residue symbols of Lagrange and Jacobi. Quadratic reciprocity.
• The solution of quadratic equations in integers.
• Introduction to one or more of the following topics, depending on time available: Diophantine approximation and transcendence, Dirichlet's theorem on primes in arithmetic progressions, Diophantine equations and elliptic curves.

Lecture Notes:

• Lecture 1: Divisibility
• Lecture 2: Primes and the Fundamental Theorem of Arithmetics
• Lecture 3: Congruences (Fermat and Euler Theorems)
• Lecture 4: Chineese reminder theorem and multiplicative functions
• Lecture 5: Applications to cryptography and computations
• Lecture 6: Congruences modulo primes
• Lecture 7: Congruences modulo prime powers
• Additional reading: p-adic numbers (non-examinable)
• Lecture 8: Primitive roots
• Lecture 9: Quadratic residues and law of quadratic reciprocity
• Lecture 10: Jacobi symbol
• Lecture 11: Arithmetic functions
• Lecture 12: Diophantine approximation
• Lecture 13: Diophantine approximation and algebraic numbers
• Lecture 14: Geometry of numbers (non-examinable)

Math Cafe: Tuesday 9-10am at Portacabin 4, run by Demmas Salim

Homework Problems:

• Problem Set 1: due Friday, Feb. 3 --- Solutions
• Problem Set 2: due Friday, Feb. 17 --- Solutions
• Problem Set 3: due Friday, March 3 --- Solutions
• Problem Set 4: due Friday, March 17 --- Solutions
• Problem Set 5: due Friday, March 31 --- Solutions
• Problem Set 6: due Friday, April 28 --- Solutions

Sample Exam:

• Exam --- Solutions

  (Now there is no 4/5 rule. To get full mark all problems have to be solved.)

References:

• Alan Baker, A concise introduction to the theory of numbers. Cambridge University Press, 1984.
• Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An introduction to the theory of numbers. Fifth edition. John Wiley \& Sons, Inc., 1991.
• H. E. Rose, A course in number theory. Second edition. Oxford Science Publications. The Clarendon Press, Oxford University Press, 1994.
• J. H. Silverman, A friendly introduction to number theory. Third edition. Prentice Hall, 2005.