ECM3704 - Number Theory (2016/2017)
Lecturer: Dr Julio Andrade
Office: Harrison 273
Email: j.c.andrade@exeter.ac.uk
Lecture times
Mondays 9:30am - 10:30am Newman Collaborative LT (C/D)
Tuesdays 3:30pm - 4:30pm Newman Collaborative LT (C/D)
Fridays 9:30am - 10:30am Harrisonn 103
Office hours: Mondays 1:30pm - 2:30pm
Tuesdays 11:30am - 12:30pm
(or by appointment - send an email)
Assignments and Assessment:
There will be 2 assessed coursework assignments, each contributing 10% of the module mark, due in by noon on the following dates:
Monday 7th November 2016
Monday 5th December 2016
There will also be several unassessed examples sheets.
There will be a 2-hour exam in the May / June contributing 80% of the module mark.
Complete Description of the Course (.pdf)
Office: Harrison 273
Email: j.c.andrade@exeter.ac.uk
Lecture times
Mondays 9:30am - 10:30am Newman Collaborative LT (C/D)
Tuesdays 3:30pm - 4:30pm Newman Collaborative LT (C/D)
Fridays 9:30am - 10:30am Harrisonn 103
Office hours: Mondays 1:30pm - 2:30pm
Tuesdays 11:30am - 12:30pm
(or by appointment - send an email)
Assignments and Assessment:
There will be 2 assessed coursework assignments, each contributing 10% of the module mark, due in by noon on the following dates:
Monday 7th November 2016
Monday 5th December 2016
There will also be several unassessed examples sheets.
There will be a 2-hour exam in the May / June contributing 80% of the module mark.
Complete Description of the Course (.pdf)
Description
Number theory is a vast and fascinating field of mathematics, consisting of the study of the properties of whole numbers. From this module, you will acquire a
working knowledge of the main concepts of classical elementary number theory, together with some appreciation of modern computational techniques.
This course covers one of the oldest and most popular areas of mathematics, building on basic ideas and including modern applications. The dual objectives are to provide a solid foundation for further work in number theory, but also at the same time to give a self-contained interesting course suitable as an end in itself, with modern answers to ancient problems and modern applications of classical ideas. You will acquire a sound foundation in number theory from a modern perspective.
Number theory is a vast and fascinating field of mathematics, consisting of the study of the properties of whole numbers. From this module, you will acquire a
working knowledge of the main concepts of classical elementary number theory, together with some appreciation of modern computational techniques.
This course covers one of the oldest and most popular areas of mathematics, building on basic ideas and including modern applications. The dual objectives are to provide a solid foundation for further work in number theory, but also at the same time to give a self-contained interesting course suitable as an end in itself, with modern answers to ancient problems and modern applications of classical ideas. You will acquire a sound foundation in number theory from a modern perspective.
Syllabus Plan
- divisibility, greatest common divisor;
- extended Euclidean algorithm, prime numbers and unique factorisation;
- congruences, Euler's and Wilson's theorems, Chinese Remainder Theorem;
- computational methods, primality testing, factorisation, RSA cryptosystem;
- primitive roots;
- quadratic residues and quadratic reciprocity;
- sums of two and four squares;
- Pythagorean triples;
- Fermat's Last Theorem for exponent four.
Synopsis
The following syllabus is tentative and subject to change.
Day Content Recommended Problems
19/09/2016 Course Rules, Motivation and Section 1.1 and Section 1.2 Problem Sheet 1: Exercise 1
20/09/2016 Section 1.2 and Section 1.3 Problem Sheet 1: Exercises 3 and 4
23/09/2016 Section 1.3 and Section 1.4
26/09/2016 Section 1.4 and Section 1.5 Problem Sheet 1: Exercise 2
27/09/2016 Section 1.6 up to Definition 1.25 Problem Sheet 1: Exercise 5
30/09/2016 Section 1.7 and Section 2.1 Problem Sheet 1: Exercises 6, 7 and 8
03/10/2016 Section 2.1 Problem Sheet 2: Exercises 1 and 2
04/10/2016 Section 2.1 and Section 2.2 Problem Sheet 2: Exercise 3
07/10/2016 Tutorial - Problem Class
10/10/2016 Section 2.2
11/10/2016 Section 2.2 and Section 2.3 Problem Sheet 2: Exercise 4 and 5
14/10/2016 Section 2.3 (including Example 2.28)
17/10/2016 Section 2.4 and Section 2.5
18/10/2016 Section 2.5 Problem Sheet 2: Exercise 7
21/10/2016 Section 2.6 (up to Corollary 2.47) Problem Sheet 2: Exercise 6
24/10/2016 Section 2.6 and Section 2.7
25/10/2016 Section 2.8 Problem Sheet 2: Exercise 8
28/10/2016 Section 2.9 and Section 2.10 (statement of Lagrange's theorem)
31/10/2016 Section 2.10
01/11/2016 Section 2.11 Problem Sheet 2: Exercise 9
04/11/2016 Section 2.11 (A different approach to Hensel's Lemma and proof of the Lemma)
07/11/2016 Section 2.12
08/11/2016 Section 2.12 and Section 2.13 Problem Sheet 3: Exercises 1, 2, 3, 4, 5
11/11/2016 Section 3.1
14/11/2016 Section 3.2 and Section 3.3 (Lecture delivered by Gihan)
15/11/2016 Section 3.3 and Section 3.4 (Lecture delivered by Valentina)
18/11/2016 Section 3.4 (including the proof of Theorem 3.21) Problem Sheet 3: Exercise 7
21/11/2016 Section 3.4 and Section 3.5 Problem Sheet 3: Exercise 6
22/11/2016 Section 3.5 and Section 3.6
25/11/2016 Section 3.6 (including the proof of Theorem 3.33) Problem Sheet 3: Exercises 8, 9 and 10
28/11/2016 Section 3.6 and Section 4.1
29/11/2016 Section 4.1 Problem Sheet 4: Exercises 7 and 8
02/12/2016 Section 4.2 (including proof of Theorem 4.10)
05/12/2016 Section 4.3 Problem Sheet 4: Exercises 1, 2, 3, 4,
06/12/2016 Section 4.4
09/12/2016 Section 4.4 (last lecture) Problem Sheet 4: Exercises 5 and 6
The following syllabus is tentative and subject to change.
Day Content Recommended Problems
19/09/2016 Course Rules, Motivation and Section 1.1 and Section 1.2 Problem Sheet 1: Exercise 1
20/09/2016 Section 1.2 and Section 1.3 Problem Sheet 1: Exercises 3 and 4
23/09/2016 Section 1.3 and Section 1.4
26/09/2016 Section 1.4 and Section 1.5 Problem Sheet 1: Exercise 2
27/09/2016 Section 1.6 up to Definition 1.25 Problem Sheet 1: Exercise 5
30/09/2016 Section 1.7 and Section 2.1 Problem Sheet 1: Exercises 6, 7 and 8
03/10/2016 Section 2.1 Problem Sheet 2: Exercises 1 and 2
04/10/2016 Section 2.1 and Section 2.2 Problem Sheet 2: Exercise 3
07/10/2016 Tutorial - Problem Class
10/10/2016 Section 2.2
11/10/2016 Section 2.2 and Section 2.3 Problem Sheet 2: Exercise 4 and 5
14/10/2016 Section 2.3 (including Example 2.28)
17/10/2016 Section 2.4 and Section 2.5
18/10/2016 Section 2.5 Problem Sheet 2: Exercise 7
21/10/2016 Section 2.6 (up to Corollary 2.47) Problem Sheet 2: Exercise 6
24/10/2016 Section 2.6 and Section 2.7
25/10/2016 Section 2.8 Problem Sheet 2: Exercise 8
28/10/2016 Section 2.9 and Section 2.10 (statement of Lagrange's theorem)
31/10/2016 Section 2.10
01/11/2016 Section 2.11 Problem Sheet 2: Exercise 9
04/11/2016 Section 2.11 (A different approach to Hensel's Lemma and proof of the Lemma)
07/11/2016 Section 2.12
08/11/2016 Section 2.12 and Section 2.13 Problem Sheet 3: Exercises 1, 2, 3, 4, 5
11/11/2016 Section 3.1
14/11/2016 Section 3.2 and Section 3.3 (Lecture delivered by Gihan)
15/11/2016 Section 3.3 and Section 3.4 (Lecture delivered by Valentina)
18/11/2016 Section 3.4 (including the proof of Theorem 3.21) Problem Sheet 3: Exercise 7
21/11/2016 Section 3.4 and Section 3.5 Problem Sheet 3: Exercise 6
22/11/2016 Section 3.5 and Section 3.6
25/11/2016 Section 3.6 (including the proof of Theorem 3.33) Problem Sheet 3: Exercises 8, 9 and 10
28/11/2016 Section 3.6 and Section 4.1
29/11/2016 Section 4.1 Problem Sheet 4: Exercises 7 and 8
02/12/2016 Section 4.2 (including proof of Theorem 4.10)
05/12/2016 Section 4.3 Problem Sheet 4: Exercises 1, 2, 3, 4,
06/12/2016 Section 4.4
09/12/2016 Section 4.4 (last lecture) Problem Sheet 4: Exercises 5 and 6
Lecture Notes
Lecture Notes
Guide to basic study skills (please read before attempting coursework)
A Different Proof of Theorem 2.8
Hensel's Lemma
Lecture Notes
Guide to basic study skills (please read before attempting coursework)
A Different Proof of Theorem 2.8
Hensel's Lemma
Problem Sheets
Problem Sheet 1 (non-assessed)
Solutions to Problem Sheet 1 (non-assessed)
Problem Sheet 2 (assessed)
Solution to Problem Sheet 2 (assessed)
Problem Sheet 3 (assessed)
Solution to Problem Sheet 3 (assessed)
Problem Sheet 4 (non-assessed)
Solution to Problem Sheet 4 (non-assessed)
Problem Sheet 1 (non-assessed)
Solutions to Problem Sheet 1 (non-assessed)
Problem Sheet 2 (assessed)
Solution to Problem Sheet 2 (assessed)
Problem Sheet 3 (assessed)
Solution to Problem Sheet 3 (assessed)
Problem Sheet 4 (non-assessed)
Solution to Problem Sheet 4 (non-assessed)
Bibliography
[1] Burn R.P.; A Pathway into Number Theory 2nd, Cambridge University Press.
[2] Niven I. & Zuckerman H.S. & Montgomery H.L.; An Introduction to the Theory of Numbers 5th, Wiley
[3] Rose H.E.; A Course in Number Theory, Oxford University Press
[4] Rosen K.H.; Elementary Number Theory and its Applications, Addison-Wesley
[1] Burn R.P.; A Pathway into Number Theory 2nd, Cambridge University Press.
[2] Niven I. & Zuckerman H.S. & Montgomery H.L.; An Introduction to the Theory of Numbers 5th, Wiley
[3] Rose H.E.; A Course in Number Theory, Oxford University Press
[4] Rosen K.H.; Elementary Number Theory and its Applications, Addison-Wesley
