## Number Theory

During the maths camp last summer we used Simon Singh's excellent website The Black Chamber after introducing the students to the concept of codebreaking by splitting them into groups and asking them to create their own codes. We looked at the frequency distributions of letters in English and Kiswahili and how you could use these to crack certain types of code.

We followed this with a session on modular arithmetic, leading to the use of primes in codebreaking, via inverses.

In the final two sessions we showed the students three problems and asked the following questions:

**Fermat’s Last Theorem **(with n=3)

x^{3}+y^{3}=z^{3}

and asked students if they could find any integer solutions.

**H****endrik Lenstra sequence**

$latex x_{0}=1$

Etc. etc.

Are the solutions always integers?

**3n+1 Problem **

Take any positive integer, if it is odd: multiply it by 3 and add 1, if it is even: half it.

Keep following these steps, stopping if you get to 1.

e.g. 5 – 16 – 8 – 4 – 2 – 1

Are there any numbers which don’t go to 1?

Students had time to explore these using pen and paper and then used Mathematica applets to explore further

We then used these problems to discuss the difference between an unsolved problem, a counter example and a proof.

See the 3n+1 it described here by Stephen Wolfram

Fermat's Last Theorem - see Simon Singh's site (the book is brilliant)

Lenstra ...