Number Theory and Geometry

This module runs in alternate years: 2012-13, 2014-15 and so on.

EMMS093S6 (30 credits)

Aims

This is a two-part course aiming to provide you with an introduction to two important areas of pure mathematics, number theory and geometry -- topics which every pure mathematician will find of interest.

The number theory section will cover types of numbers such as polygonal numbers and perfect numbers, followed by number theoretic functions, including Euler's φ function. We will prove Fermat's little theorem and study quadratic congruences as well as Pythagorean triples and sums of squares.

The section on geometry will devote time to vector geometry, affine geometry and Euclidean geometry. Curves arising from conic sections, such as the ellipse and the hyperbola, will also be studied and their properties derived from first principles, with some applications and generalisations. Finally there will be a look at the geometry of the complex plane.

Teaching and Assessment

Teaching for this module will take place throughout the year, with eight evenings in each of the Autumn and Spring Terms and two evenings of revision and consolidation in the Summer Term.

Of the final course mark, 80% is based on a three-hour exam in June and the other 20% is from assessed coursework.

Coursework  will consist of short, problem based assignments. You will have around three weeks to complete each assignment. The examination in the Summer Term has three sections. Section A (worth 40%) consists of compulsory short questions. Sections B and C (worth 20% each) contain several longer questions. You must answer one from Section B and one from Section C.

Syllabus

Number Theory

• Numbers with names: polygonal numbers, Fermat primes, Mersenne primes, perfect numbers.
• Number theoretic functions, including Euler's φ function.
• Fermat's little theorem and applications to public-key cryptography.
• Quadratic Congruences. Pythagorean triples and sums of squares.

(8 weeks/ 24 hrs)

Geometry

• Vectors; affine transformations; dot products; Euclidean geometry; some basic theorems.
• Conics: Circles, parabolas, ellipses, hyperbolas; equations of conics; properties of conics; applications; generalisations.
• Geometry in the Complex Plane: Lines and Circles; the extended complex plane; Möbius transformations.

(8 weeks/ 24 hrs)

In addition there will be 2 weeks/6 hours of consolidation and revision lectures in the Summer Term.

Learning Outcomes

Subject Specific

1. Knowledge and understanding of, and the ability to use, mathematical and/or statistical methods and techniques.
In particular:

• Understanding of the importance of quadratic residues in the solution of congruences.
• Ability to work with the geometry of the complex plane including the effect of Mobius transformations.

2. Knowledge and understanding of a range of results in mathematics and/or statistics.
In particular:

• Familiarity with various named types of numbers, such as polygonal numbers, Mersenne primes and perfect numbers.
• Familiarity with Euler’s φ function and other number theoretic functions.
• Awareness of the usefulness of vector methods in geometry.
• Knowledge of the different types of conics; their equations and properties.

3. Appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments.

4. Appreciation of the power of generalization and abstraction in the development of mathematical theories.

5. A deeper knowledge of some particular areas of mathematics and/or statistics.
In particular:

• number theory, Euclidean geometry and geometry of the complex plane.

Intellectual

1. Ability to comprehend conceptual and abstract material.

2. Develop a logical and systematic approach to problem solving.

Practical

1. Problem-solving skills, including the ability to assess problems logically and approach them analytically.

2. Highly developed quantitative skills.

3. Ability to transfer knowledge and expertise from one context to another.
In particular:

• Awareness of some of the applications of pure mathematics, such as public key cryptography.
• Ability to analyse more general congruences extending the work done on linear congruences in the first year.
• Appreciation that there is a continuity of mathematical thought from Euclid to the present day.

Personal and Social

1. Ability to work independently with patience and persistence.

2. Time-management skills and organizational skills.

3. Good communication skills, including the ability to write coherently.

4. Ability to complete work in a limited time period.

Indicative Reading List

• Jones, Gareth A and J Mary Jones, Elementary Number Theory, Springer-Verlag Berlin (1998).
• James J Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press(1999).
• Roger A Fenn, Geometry, Springer-Verlag (2001).
• John Roe, Elementary Geometry, Clarendon Press (1993).
• Priestley, HA, An introduction to Complex Analysis, Oxford University Press (2003).