A Gentle Proof of Fermat's Little Theorem and Fermat-Euler Theorem

In this video, we dive deep into the proof of Fermat's Little Theorem using the necklace method and Fermat-Euler Theorem using number and group theory.

Fermat's Little Theorem states that if p is a prime number and a is any integer not divisible by p, then a raised to the power of p-1 is congruent to 1 modulo p. This theorem is a cornerstone of number theory and has important implications in cryptography.

We also explore the Fermat-Euler Theorem, which is a generalization of Fermat's Little Theorem. The theorem states that if a and n are coprime integers, then a raised to the power of phi(n) is congruent to 1 modulo n, where phi(n) is Euler's totient function.

We explain these theorems using the necklace method and number and group theory, and we highlight the importance of these concepts in modern cryptography. Specifically, we show how the RSA cryptosystem depends on these theorems.

If you're interested in the world of number theory and cryptography, and want to learn more about the proofs of Fermat's Little Theorem and Fermat-Euler Theorem, then this video is for you.

Tutorials:

Contents of this video
00:00 - Introduction
01:02 - Proof of Fermat's Little Theorem
04:38 - Fermat's Little Theorem vs Euler's Theorem
05:35 - Euler's Totient Function
07:45 - Fermat-Euler Theorem

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[samuelmanni]

I think, you should also mention, that ax_i \neq ax_j, for i \neq j. This is the reason, why there is a 1-1 function between K and S. The property is true, since if not, then ax_i-ax_j = 0 mod n, but then x_i-x_j = 0 (since a coprime to n) and therefore x_i = x_j, which is a contradiction. Since all the elements in K are coprime to n and different, there are as many elements in K as in S, and there is a 1-1 function between the two sets.