Every week we dedicate an hour to Beautiful Mathematics – the Mathematics that shows us how Beautiful is our Intellect. Today we are going to discuss the Fermat’s Little Theorem.
This week, I decided to do three beautiful proofs in this one-hour session…
- Proof of Fermat’s Little Theorem ( via Combinatorics )
It uses elementary counting principles. Here is a Wikipedia resource to learn it.
- Fibonacci Tilling
This is about how we will prove that the number of ways we can cover a 2xn board with a 2×1 dominoes is the same as – the n-th Fibonacci Number.
We will also discuss some identities related to the Fibonacci Numbers using this idea of the Tiling.
For eg: .
- Sylvester – Gallai Theorem and Extremal Principle
Here, we will show the proof of the theorem which states that
The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either
- all the points lie on a single line; or
- there is at least one line which contains exactly two of the points
This will introduce the students to the idea of the Extremal Principle and the Well Ordering Principle.