A Gentle Proof of Fermat's Little Theorem by Induction

What do math, cryptography and magic have in common?🤔 Find this out in my latest YouTube video, where I unveil the secret behind Fermat's Little Theorem using an inductive proof🔍 Hint: this is the magic that makes RSA encryption work.

In this video, we explore the proof of Fermat's Little Theorem using the technique of mathematical induction. Fermat's Little Theorem is a fundamental theorem in number theory that states that for any prime number p and any integer a not divisible by p, a raised to the power of p-1 is congruent to 1 modulo p. Euler's generalization of Fermat's Little Theorem, which extends this result to any positive integer n, is the backbone of the RSA algorithm, one of the most widely used encryption methods in modern cryptography. We'll begin by discussing the basics of modular arithmetic, then dive into the proof by induction of this important theorem while we are using binomial theorem, binomial expansion, pascal's triangle and necklace method. Whether you're a student studying number theory or just curious about the math behind it, this video will provide an accessible and engaging overview of Fermat's Little Theorem and its proof, as well as its connection to the RSA algorithm.

Contents of this video:
00:00 - Introduction
00:53 - Fermat's Little Theorem vs Euler's Theorem
01:30 - Inductive Proof Definition
03:11 - Expanding the theorem with Binomial Theorem
05:57 - Binomial Expansion and Pascal's Triangle
07:35 - Necklace method

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[BulentSiyah]

I had no idea that number theory and cryptography had such a strong connection. The way this video breaks down the proof of Fermat's Little Theorem using binomial theorem and Pascal's triangle is really impressive.