The aim of this module is to introduce students to some of the basic ideas of number theory, and to use this as a context in which to discuss the development of mathematics through examples, conjectures, theorems, proofs and applications. The module will introduce and illustrate different methods of proof in the context of elementary number theory, and will apply some basic techniques of number theory to cryptography.
One of the pre-requisites for MATH3078
Aims and Objectives
Having successfully completed this module you will be able to:
- state and prove Fermat's Little Theorem & its generalisation using Euler's function & use them to implement the RSA cipher & dicrete log cipher
- apply different methods of proof to verify mathematical assertions, including proof by induction, by contrapositive and by contradiction
- analyse hypotheses and conclusions of mathematical statements
- solve systems of Diophantine equations using the Chinese Remainder Theorem & the Euclidean algorithm
- understand the basics of modular arithmetic
- Proof and Mathematical Logic
- Number Theory
- Divisibility, least common multiples, Euclid's algorithm.
- Integer solutions of ax + by = c.
- Prime numbers and prime-power factorisations, irrational numbers.
- Existence of infinitely many primes.
- Modular arithmetic, linear congruences. Chinese Remainder Theorem.
- Fermat's Little Theorem.
- Units, Euler's function, Euler’s Theorem.
- Diffie-Hellman-Merkle and Rivest-Shamir-Adleman key exchange systems
- Pythagorean triples & Fermat’s Last Theorem.
Learning and Teaching
Teaching and learning methods
Lectures, problem classes, workshops, private study
|Wider reading or practice||10|
|Completion of assessment task||20|
|Preparation for scheduled sessions||12|
|Supervised time in studio/workshop||6|
|Total study time||150|
Resources & Reading list
ROSEN K H (1988). Number Theory and its Applications. Addison-Wesley.
JONES G A & JONES J M (1998). Elementary Number Theory.. Springer.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External