Developing Ideas in Number Theory Report

This document outlines a report assignment for the 'Developing Ideas in Number Theory' module. Students are expected to explore topics from the secondary math syllabus, such as primes, factors, and equations, and present a substantial report on one or two topics. The assignment requires students to develop and modify conjectures, create their own examples, and potentially refer to readings in the area. Students will attend two tutorials to receive guidance on their work. The report should demonstrate a competent use of number theory techniques, clear communication, and the ability to explore and prove conjectures. The document also includes a reading list and a schedule of sessions and tutorials.

Developing Ideas in Number
Theory Module Handbook
77-6744-00S
2016 Semester 1
Module leader: Jo Tomalin
Level 6

Introduction
In this module we will be looking in more depth at ideas that occur
throughout the secondary maths syllabus, such as primes, factors, fractions,
division, equations.
You will be working in groups on exploring situations, developing and
modifying conjectures and looking for proofs of different kinds.
We will use ICT in some sessions, particularly Excel to aid our explorations.
You will be expected to do plenty of collaborative and individual work
outside the taught sessions, on consolidating and completing class work
during the taught sessions, and then on developing ideas towards your
assignment.
You should be able to come to your first tutorial with a substantial amount of
work to discuss. And you should expect to use both of your tutorials to take
you further with the work than you could get on your own.

Assignment
Present a substantial report on one or two topics that you have explored yourself, taking as
starting points the work explored in class and suggested activities.
This work should include plenty of your own exploration. It may (but does not have to) refer to
some reading in the area. (A Pathway into Number Theory by R P Burns is a possible starting point
for some investigations).
It should include explorations where you explore your own conjectures, and create your own
examples.
You may well find that your work develops in surprising directions away from your starting point.
You are expected to attend two tutorials and follow your tutor's advice on directions to develop
your work in.
Assessment Criteria
1 Makes use of ideas and techniques from the area of number theory competently. (This
may include appropriate use of ICT.)
2 Makes conjectures, verify , refute and modify them, and sometimes prove them.
3 Creates own examples to explore ideas and techniques further (which may be
developed from reading).
4 Communicates clearly in words, symbols, and, where appropriate, diagrams.
You will have two twenty minute tutorials. It is important that you bring substantial amounts of your
own work to both tutorials.
It is suggested that you approach several topics, but you may choose to hand in only one, if you feel
you can do a substantial amount of work which fits all the assessment criteria.
A rough guide to the amount of work expected, is between 10 and 20 sides of A4 paper. (A 20 credit
level 6 assignment would normally be 5000 words, but words are not a very useful guide to work like
this which includes algebra and diagrams.)
All reading should be APA referenced, and all collaborations with others or discussions with tutors
also referenced.
If you work with others (which may be an excellent thing) it would be sensible to try then to develop
your own independent examples on the topics you have worked on.

Reading List
Burn R.P (1996) A Pathway into Number Theory Cambridge University Press
This is an excellent book to use to immerse yourself in a particular topic, as the book works by taking
you through examples. If you use the book to understand these and the ideas behind them, you
should then be able to develop your own similar examples.
Burn R.P and Chetwynd A (1995) A Cascade of Numbers: An Introduction to Number Theory
Butterworth-Heinemann
This is a more accessible book, also presented as activities for you to do. You may find it helpful with
some ideas.
Stark H (1978) An Introduction to Number Theory MIT Press
Davenport (1952) The Higher Arithmetic: An Introduction to the Theory of Numbers Cambridge
University Press
Burton (2006) Elementary Number Theory, Allyn & Bacon
Goldman, J. (1998) The Queen of Mathematics, A K Peters
These four books are a selection of books that cover some of the topics we look at. You may also
want to search the library shelves for books on number the
Hardy G H and Wright E.M (2008) An Introducton to the Theory of Numbers 6th Edition Oxford
University Press
(This is a recent edition of a classic book, with updates in the chapters. It is a challenging book, and
you will find most of it beyond you, but you may find some things of interest here.)
Some Background Reading
Conway, J. (2006) The Book of Numbers, Copernicus Books
Aczel, A. (1997) Fermat's Last Theorem, Penguin
Hoffman, P. (1999) The Man Who Loved Only Numbers, Fourth Estate
Advanced spread sheet software e.g. EXCEL
Programming language, e. g. Logo, Visual Basic

Sessions
Week
number
Week
begins
Tuesday pm
confirmed
Friday am confirmed
Mon 5
8 Mon 12 Taught Session Taught Session
9 Mon 19 Sep Taught Session Taught Session
10 Mon 26 Sep Taught Session Taught Session
11 Mon 3 Oct Taught Session Taught Session
12 Mon 10 Oct Taught Session Taught Session
13 Mon 17 Oct Taught Session Taught Session
14
Mon 24 Oct
(half-term?
check)
Reading Week
15 Mon 31 Oct Study Session Study Session
16 Mon 7 Nov Study session Study session
17 Mon 14 Nov First Tutorials 1-
2.40
First Tutorials 10.40-
12
18 Mon 21 Nov First Tutorials 1-
2.40
First Tutorials 10.40-
12
19 Mon 28 Nov Study session Study session
20 Mon 5 Dec Study session Study session
21 Mon 12 Dec Second Tutorials 1-
2.40
Second Tutorials
10.40-12
22*** Mon 19 Dec Holiday23*** Mon 25 Dec
24
Mon 2nd Jan
(term starts
Tuesday)
Second Tutorials1-
2.40
Second Tutorials
10.40-12 (Wed am)
25 Mon 9 Jan Assignment hand-in 3pm Friday 13th January