Module manager: Dr Vincenzo L Mantova
Email: V.L.Mantova@leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2024/25
MATH2020 or MATH2022 or equivalent
This module is approved as a discovery module
This module is mainly about the work of the 18th Century mathematicians Euler, Lagrange and Gauss, including such highlights as Lagrange's Theorem that every positive integer is a sum of at most four squares, and Gauss's Law of quadratic reciprocity. We shall also introduce continued fractions to help solve Pell's equation.
To introduce some of the main results and methods of elementary number theory.
On completion of this module, students should be able to:
a) work with divisors, primes and prime factorizations, and use the Euclidean algorithm;
b) compute with congruences, including using Fermat's and Euler's theorems;
c) use primitive roots and other methods to test numbers for primality;
d) calculate Legendre symbols using quadratic reciprocity and other methods;
e) use continued fractions to solve Pell's equation and to approximate reals by rationals.
- Prime factorization and applications.
- Congruences.
- Fermat's Little Theorem and its use in looking for prime factors.
- Euler's function. Wilson's Theorem.
- Pythagorean triples.
- Integers which are sums of 2,3,4 squares.
- Fermat's conjecture for Primitive roots.
- Quadratic reciprocity and applications.
- Gaussian integers and various generalisations.
- Use in solving certain Diophantine equations.
- Continued fractions.
- 'Best' approximation of reals by rationals. Pell's equation.
- Brief explanation of the principles behind public key cryptography.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 33 | 1 | 33 |
| Private study hours | 117 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 150 | ||
Studying and revising of course material.
Completing of assignments and assessments.
Regular problem solving assignments
| Exam type | Exam duration | % of formal assessment |
|---|---|---|
| Standard exam (closed essays, MCQs etc) | 2.0 Hrs 30 Mins | 100 |
| Total percentage (Assessment Exams) | 100 | |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
The reading list is available from the Library website
Last updated: 4/29/2024
Errors, omissions, failed links etc should be notified to the Catalogue Team