Zsigmondy's Theorem

The story line that guides us is proving a theorem of Zsigmondy in number theory and seeing how it can be used to solve maths olympiad problems that would otherwise be quite difficult.

To achieve this goal we first understand what I consider to be the most central topic in high school algebra which is omitted in high schools: cyclotomic polynomials. This sounds specialised but this is at the heart of all the algebra learned at high school such as factorising a difference of 2 squares or cubes. The cyclotomic polynomials gives a factorisation of x^n-1. When n is 2, this is just the difference of 2 squares. If you let the x be x/y then you really get x^2-y^2 after some easy manipulation.  (x^n means x to the power of n)

These lessons will be a very valuable part of a serious high school maths student or olympian.

One of the really interesting features of this course is that the instructor learns the proof of the Zsigmondy Theorem with the students and you get to see how to educate yourself without further need to be taught.