F18PA - Pure Mathematics A

Course leader(s):

Lotte Hollands (Edinburgh, January)

Aims

To offer an introduction to the ideas of number theory and geometry to students specialising in Mathematics.

To exploit technical know-how gained at SCQF level 7 to develop theoretical ideas in a concrete setting.

To offer further insights into Mathematical reasoning and the art of proof.

Syllabus

1. The integers (1.1 1. Well-ordering, 1.2 2. Divisibility, 1.3 3. Perfect numbers, 1.4 4. The division algorithm, 1.5 5. The Euclidean algorithm, 1.6 6. Linear Diophantine equations)

2. Prime numbers (2.1 1. Sieve of Erasthostenes, 2.2 2. The infinitude of primes, 2.3 3. Least common multiple, 2.4 4. Mersenne and Fermat primes , 2.5 5. The fundamental theorem of arithmetic)

3. Congruences (3.1 1. Congruence relations, 3.2 2. Calculations with congruences, 3.3 3. Decimal representations, 3.4 4. Solving congruence relations, 3.5 5. The multiplicative inverse, 3.6 6. The Chinese remainder theorem)

4. Multiplicative functions (4.1 1. Multiplicative functions , 4.2 2. The Euler totient function, 4.3 3. Mobius inversion, 4.4 4. Fermat's little theorem, 4.5 5. Cryptography)

Learning outcomes

By the end of the course, students should be able to do the following:

Analyse key properties of the integers, such as well-ordering and divisibility, by for instance deriving rigorous statements about prime and perfect numbers.
Employ the Euclidean algorithm and magic matrix method to find all solutions to a linear Diophantine equation.
Formulate the fundamental theorem of arithmetic and reproduce Euler’s beautiful proof of the infinitude of prime numbers.
Prove arithmetic properties of congruences, as well as systematically solve systems of congruence equations, possibly by invoking the Chinese remainder theorem.
Compute with multiplicative functions, such as Euler’s totient function, applying for instance Mobius inversion, Euler’s theorem and Fermat’s little theorem.
Work with the RSA encryption protocol, implementing the newly acquired knowledge about congruences and multiplicative functions.

Further details

Curriculum explorer: Click here

SCQF Level: 8

Credits: 15