## MAS330 Topics in Number Theory

 Semester 1, 2019/20 10 Credits Lecturer: Dr Evgeny Shinder Home page (also MOLE) Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

The course covers topics in elementary Number Theory. This includes Modular Arithmetic, and properties of primes and integers. Most of the material (with the notable exception of the RSA cryptosystem) has been introduced by Fermat, Euler and Gauss in the 17th, 18th and 19th centuries.

Prerequisites: MAS211 (Advanced Calculus and Linear Algebra)

The following modules have this module as a prerequisite:

 MAS345 Codes and Cryptography

## Outline syllabus

1. Modular Arithmetic
• Linear Congruences
• Fermat's Little Theorem and the RSA cryptosystem.
• Arithmetic functions.
• Euler's function and Euler's theorem.

2. Primes, Integers and Equations
• Perfect numbers, Mersenne primes, Fermat primes.
• Pythagorean triples and Fermat Last Theorem.
• Generating functions and partitions.
• Continued fractions.
• Pell's equation.

## Aims

• To introduce various topics in number theory

## Learning outcomes

By the end of the modules students should be able to: - Solve linear and quadratic congruences Know and use Fermat's Little Theorem and Wilson's Theorem Code and decode numbers in the RSA cryptosystem Determine whether a number is a quadratic residue modulo an odd prime using the Law of Quadratic Reciprocity. Show that there are infinitely many primes of certain types Verify that certain Mersenne numbers are prime Know Euclid's description of an even perfect number Work out all Pythagorean triples containing a given number Know the statement of Fermat's Last Theorem and something of its history Express a rational number as a finite continued fraction and hence solve a linear diophantine equation Express a given repeated continued fraction in terms of a surd Expand a surd as an infinite continued fraction and hence find a convergent which is an approximation to the given surd to a given degree of accuracy Solve a Pell equation from a continued fraction expansion

## Teaching methods

Lectures, problem sheets. It is highly recommended to work on the problems and submit the assigned problems which will be marked for feedback.

20 lectures, no tutorials

## Assessment

Assessment: one formal 2 hour and 30 minutes written examination (4 questions out of 4). On the exam you will be solving problems using methods we learnt in class. You will also need to be able to give definitions and state theorems of the course, and to apply these theorems creatively.

Note: calculators are not allowed on the exam.

## Full syllabus

Week 1: Linear Congruences
Fundamental theorem of arithmetic; Euclidean algorithm; review of modular arithmetic; solving linear congruences with Euclidean algorithm; Chinese remainder theorem.

Week 2: Fermat's Little Theorem, The RSA cryptosystem.
Fermat's Little Theorem, the RSA algorithm.
Week 3: Arithmetic functions.
The functions τ and σ; multiplicativity of τ and σ; the Möbius inversion formula.
Week 4: Euler's function and Euler's theorem.
Euler's ϕ-function; multiplictivity of ϕ; Euler's theorem; quadratic residues and non-residues; Euler's criterion;.
Week 5: Gauss' Quadratic Reciprocity Law.
Week 6: Perfect numbers, Mersenne primes, Fermat primes.
Formula for even perfect numbers; Mersenne primes, Fermat primes.
Week 8: Pythagorean triples and Fermat Last Theorem.
Finding all solutions of x2 + y2 = z2; statement of the Fermat Last Theorem; equation m = x2 + y2.
Week 9: Generating series and partitions.
Generating series. Partitions.
Week 10: Continued fractions.
Representing rational numbers as a finite continued fraction; the convergents; numbers pk, qk; ratio of two consequtive Fibonacci numbers; representing irrational numbers as infinite continued fractions; periodic continued fractions.
Week 11: Pell's equation.
Solving Pell's equation using continued fractions; the fundamental solution; all solutions may be obtained from the fundamental solution.