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## ECM3704 - Number Theory (2018)

MODULE TITLE | Number Theory | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECM3704 | MODULE CONVENER | Dr Henri Johnston (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 11 weeks | 0 | 0 |

Number of Students Taking Module (anticipated) | 142 |
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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: ECM1706 or equivalent

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.

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.

- 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;

- quadratic residues and quadratic reciprocity;

- sums of two and four squares;

- Pythagorean triples;

- Fermat's Last Theorem for exponent four.

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad | 0.00 |
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Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures/example classes |

Guided independent study | 117 | Guided independent study |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Not applicable | |||

Coursework | 20 | Written Exams | 80 | Practical Exams |
---|

Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book | 80 | 2 hours - Summer Exam Period | All | Upon request |

Coursework – example sheets | 20 | Variable | All | Written and verbal |

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 |

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.

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 | 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 | Rose H.E. | A Course in Number Theory | Oxford University Press | 1994 | 000-0-198-53261-X | [Library] | |

Set | Rosen K.H. | Elementary Number Theory and its Applications | Addison-Wesley | 2005 | 000-0-201-57889-1 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | ECM1706 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 6 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Thursday 06 July 2017 | LAST REVISION DATE | Wednesday 27 February 2019 |

KEY WORDS SEARCH | Number theory; prime numbers; divisibility; quadratric reciprocity; congruences; sums of squares; crytography. |
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