MA220-5-SP-CO:
Number Theory
2021/22
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Spring
Undergraduate: Level 5
Current
Monday 17 January 2022
Friday 25 March 2022
15
12 May 2021
Requisites for this module
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This module gives a broad introduction to the mathematics of the integers, primes and modular arithmetic, including both classical and modern viewpoints.
Number theory encompasses some of the most classical and important topics in mathematics, stemming from the study of integers, Diophantine equations, prime numbers and modular arithmetic. As well as introducing each of these, in this module it will be demonstrated how techniques from a range of mathematical disciplines such as algebra and geometry can be applied to these topics.
On completing this module, students should:
1. Understand a range of mathematical tools relating to Diophantine equations and associated mathematics.
2. Understand well-known properties of modular arithmetic such as the Chinese Remainder Theorem.
3. Be able to perform routine calculations in number systems such as the Gaussian integers.
4. Understand the definitions and basic properties of algebraic and transcendental numbers and continued fractions.
5. Have an awareness of famous open problems and modern avenues of research.
Syllabus:
- Introduction to Diophantine equations – Pythagorean triples, solutions via points on rational curves.
- Integer and modular arithmetic – Fundamental Theorem of Arithmetic; applications of the Euclidean algorithm; modular arithmetic; Chinese Remainder Theorem; the Euler phi function and its properties.
- Quadratic reciprocity – Quadratic residues; Euler's criterion; relation to sums of squares.
- Gaussian integers – The norm and its properties; Gaussian primes; sums of squares; related number rings.
- Diophantine approximation and continued fractions – algebraic and transcendental numbers, Diophantine approximation, solution to Pell's equation.
- Asymptotics and distribution of primes – asymptotic notation, statement of the Prime Number Theorem.
Three contact hours per week. This will consist mainly of lectures, with one class every one or two weeks. Three revision lectures to be given in the summer term.
This module does not appear to have a published bibliography for this year.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Assignment 1 |
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| Coursework |
Assignment 2 |
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| Exam |
Main exam: 180 minutes during Summer (Main Period)
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Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Alastair Litterick, email: a.litterick@essex.ac.uk.
Dr Alastair Litterick
a.litterick@essex.ac.uk
Yes
No
Yes
Prof Stephen Langdon
Brunel University London
Professor
Available via Moodle
Of 1826 hours, 30 (1.6%) hours available to students:
1796 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).
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