Number Theory - MAT00085H
Module summary
Number theory utilises an unusually wide variety of methods and proofs, many of which are surprisingly deep. This course will introduce some of the fundamental problems in this subject, ranging from finding the most accurate approximation to real numbers with continued fractions (think π and 22/7), to providing a deeper and more quantitative understanding of the structure of the real numbers through Diophantine approximation. This module will illustrate the interplay of different branches of mathematics and basic applications of number theory to cryptography.
Related modules
Additional information
Pre-requisite Modules
Introduction to Pure Mathematics
Measure & Integration for M-level
M-level version of Number Theory cannot be taken if H-level was taken previously.
Module will run
| Occurrence | Teaching period |
|---|---|
| A | Semester 1 2025-26 |
Module aims
Number theory utilises an unusually wide variety of methods and proofs, many of which are surprisingly deep. This course will introduce some of the fundamental problems in this subject, ranging from finding the most accurate approximation to real numbers with continued fractions (think π and 22/7), to providing a deeper and more quantitative understanding of the structure of the real numbers through Diophantine approximation. This module will illustrate the interplay of different branches of mathematics and basic applications of number theory to cryptography.
Module learning outcomes
By the end of the module, students should be able to:
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Demonstrate facility with the unusually wide variety of methods and proofs which appear in number theory.
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Apply algorithms taught in the course in specific calculations.
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Use a range of ideas and techniques in Diophantine approximation.
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Use algebraic and probabilistic ideas in the context of number theory.
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Use the concept of fractal dimension in the context of number theory.
Module content
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Theory of congruences.
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Sum of squares
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Elements of Cryptography
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Continued fractions and Pell’s equations
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Dirichlet’s approximation theorem and multidimensional Diophantine approximation.
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Zero-one laws and Khintchine’s theorem.
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Fractals in Number Theory
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Special assessment rules
None
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Module feedback
Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
Indicative reading
Rose, A Course in Number Theory, Oxford University Press.
G H Hardy and E M Wright, The Theory of Numbers, Oxford University Press.
Victor Beresnevich, Felipe Ramírez and Sanju Velani, Metric Diophantine Approximation: Aspects of Recent Work, Cambridge University Press