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October 10, 2019

An Hour of Beautiful Proofs

fermat's little theorem

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 2x1 dominoes is the same as F_n - the n-th Fibonacci Number.
    We will also discuss some identities related to the Fibonacci Numbers using this idea of the Tiling.

    For eg: F_0 + F_1 + ... + F_n = F_{n+2} - 1.
  • 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

  1. all the points lie on a single line; or
  2. 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.

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