MA25710 Introduction to Number Theory
Introductory description
Number theory is an ancient and beautiful subject that investigates the properties of the integers, and other questions motivated by them. For example, one studies questions like:

do equations have integer solutions? If so, do they have infinitely many integer solutions? How are those solutions distributed, and can we parametrise them? Pythagorean triples, and Fermat's Last Theorem, fall into this general area of Diophantine equations.

how do special subsets of the integers behave? The primes are probably the best example of a special subset, and one wants to know: are there infinitely many? How are they distributed? Are there interesting patterns amongst them? The most famous unsolved problem in mathematics, the Riemann Hypothesis, is connected with this.
In this module we will explore various topics that underlie the deeper study of the integers, and see some initial applications. In particular, we study:
 factorisation in the integers and in other rings
 congruences and arithmetic mod n, including primitive roots
 quadratic reciprocity
 Diophantine equations, including writing integers as sums of squares
 more advanced topics (e.g. encryption, primality checking, basic analytic prime number theory)
Module aims
To introduce students to elementary number theory and provide a firm foundation for later number theory and algebra modules.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Factorisation, divisibility, Euclidean Algorithm, Chinese Remainder Theorem.
 Congruences. Structure on Z/mZ and its multiplicative group. Theorems of Fermat and
Euler. Primitive roots.  Quadratic reciprocity, Diophantine equations.
 Elementary factorization algorithms.
 Introduction to Cryptography.
 Geometry of numbers, sum of two and four squares.
Learning outcomes
By the end of the module, students should be able to:
 Work with prime factorisations of integers
 Solve congruence conditions on integers
 Determine whether an integer is a quadratic residue modulo another integer
 Apply geometry of numbers methods to solve some Diophantine equations
 Follow advanced courses on number theory in the third year
Indicative reading list
H. Davenport, The Higher Arithmetic, Cambridge University Press.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1979.
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, SpringerVerlag, 1990.
Subject specific skills
By the end of the module the student should be able to:
 work with prime factorisations of integers
 solve congruence conditions on integers
 determine whether an integer is a quadratic residue modulo another integer
 apply geometry of numbers methods to solve some Diophantine equations
 follow advanced courses on number theory in the third and fourth year
Transferable skills
The module will help to develop skills in understanding, assessing and constructing logical arguments (especially of a quantitative nature), and presenting these clearly in writing.
Some parts of the module will explore the difference between a theoretical solution of a problem and a solution that can be practically implemented with current computing resources, a distinction that is crucial in many real world applications of mathematical concepts.
Study time
Type  Required 

Lectures  20 sessions of 1 hour (20%) 
Tutorials  9 sessions of 1 hour (9%) 
Private study  71 hours (71%) 
Total  100 hours 
Private study description
Review lectured material and work on set exercises.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Assignment  15%  
Examination  85%  

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  

Feedback on assessment
Support Classes
Marked homework will be returned to students.
Exam feedback.
Courses
This module is Core option list A for:

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Core option list B for:
 Year 3 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Core option list D for:
 Year 4 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Option list A for:

UCSAG4G1 Undergraduate Discrete Mathematics
 Year 2 of G4G1 Discrete Mathematics
 Year 2 of G4G1 Discrete Mathematics
 Year 2 of UCSAG4G3 Undergraduate Discrete Mathematics

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 2 of G105 Mathematics (MMath) with Intercalated Year
 Year 4 of G105 Mathematics (MMath) with Intercalated Year

UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of G100 Mathematics
 Year 2 of G103 Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)
 Year 3 of G100 Mathematics
 Year 3 of G103 Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)

USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 2 of GG14 Mathematics and Statistics
 Year 2 of GG14 Mathematics and Statistics

UMAAG101 Undergraduate Mathematics with Intercalated Year
 Year 2 of G101 Mathematics with Intercalated Year
 Year 4 of G101 Mathematics with Intercalated Year
This module is Option list B for:

UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 2 of GF13 Mathematics and Physics
 Year 2 of GF13 Mathematics and Physics

UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 2 of FG31 Mathematics and Physics (MMathPhys)
 Year 2 of FG31 Mathematics and Physics (MMathPhys)
 Year 3 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)

USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
 Year 2 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 2 of Y602 Mathematics,Operational Research,Stats,Economics