Mathematics

MTH3004 - Number Theory (2019)

MODULE TITLE CREDIT VALUE Number Theory 15 MTH3004 Dr Henri Johnston (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 weeks 0 0
 Number of Students Taking Module (anticipated) 86
DESCRIPTION - summary of the module content

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.

Prerequisite module: MTH1001 Structures, or equivalent

AIMS - intentions of the module

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.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 demonstrate a working knowledge of the main concepts of classical elementary number theory, together with some appreciation of modern computational techniques.

Discipline Specific Skills and Knowledge:

2 understand the role of Number Theory as a central topic in mathematics, and demonstrate an awareness of some of its modern applications;

3 comprehend a couple of useful techniques for the computation of integrals with complex methods.

Personal and Key Transferable/ Employment Skills and  Knowledge:

4 show enhanced problem-solving skills and ability to formulate your solutions as mathematical proofs;

5 reveal a fundamental knowledge of Number Theory from a modern perspective.

SYLLABUS PLAN - summary of the structure and academic content of the module

- 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;

- sums of two and four squares;

- Pythagorean triples;

- Fermat's Last Theorem for exponent four.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled Learning and Teaching Activities 33 Lectures/example classes Guided Independent Study 117 Guided independent study

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework – example sheets Variable All Written and verbal

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework – based on questions submitted for assessment 20 2 assignments, 30 hours total All Annotated script and written/verbal feedback
Written Exam – closed book 80 2 hours All Written/verbal on request, SRS

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written Exam (100%) All August Ref/Def Period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Rose H.E. A Course in Number Theory Oxford University Press 1994 000-0-198-53261-X [Library]
Set Burn R.P. A Pathway into Number Theory 2nd Cambridge University Press 1997 000-0-521-57540-0 [Library]
Set Niven I. & Zuckerman H.S. & Montgomery H.L. An Introduction to the Theory of Numbers 5th Wiley 1991 000-0-471-54600-3 [Library]
Set Rosen K.H. Elementary Number Theory and its Applications Addison-Wesley 2005 000-0-201-57889-1 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH1001
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 6 No Tuesday 10 July 2018 Friday 30 August 2019
KEY WORDS SEARCH Number theory; prime numbers; divisibility; quadratric reciprocity; congruences; sums of squares; crytography.