# MA3J9 Content

**Content:**

The module will cover several topics each year. Below is a list of possible topics:

- Sample Topic 1: Fermat's little theorem and RSA Cryptography
- Residue classes modulo primes, Fermat's little theorem, Cryptographic applications. May include Elliptic Curve factorisation
- Sample Topic 2: Hilbert's 10th problem and Undecidability
- Decidability, recursively enumerable set and Diophantine sets, Computing and algorithms
- Sample Topic 3: Hilbert's 3rd problem and Dehn invariants
- Scissor congruence in the plane, Scissor congruence in R^n and Hilbert's 3rd problem, Dehn invariant for R^3
- Sample Topic 4: Four colour theorem
- Graphs, colourings, Five colour theorem, the role of computers

** ****Aims: **

To show how a range of problems both theoretical and applied can be modelled mathematically and solved using tools discussed in core modules from years 1 and 2.

**Objectives:**

By the end of the module the student should be able to:

- For each of the topics discussed appreciate their importance in the historical context, and why mathematicians at the time were interested in it.
- For each of the topics discussed understand the underlying theory and statement of the result, and where applicable how the proof has been developed (or how a proof has been attempted in the case of unsolved problems).
- For each of the topics discussed understand how to apply the theory to similar problems/situations (where applicable).
- For each of the topics discussed understand the connections between the results/proofs in question and the core mathematics modules that the student has studied.

**Books:
**

Depending on the topics, different sources will be used. Most will be available online or with provided lecture notes.