Hello everyone,

I did like to give a brief overview of Fermat's theorem and its proof. There are various methods to prove Fermat's Little theorem, but I found the combinatorial approach to be the most straightforward and easy to understand. I'd like to discuss Fermat's theorem and its proof using combinatorics.

Fermat's Little Theorem:

It states that given 2 integers $$$a$$$ , $$$p$$$ where $$$a > 1$$$ and $$$p$$$ is a prime, It follows that $$$a^{p-1} \equiv 1 \pmod{p}$$$

Example:

Say a = 2 , p = 5. $$$a^{p-1} = 2^{5-1} = 2^4 = 16 \equiv 1 \pmod{5}$$$

Proof:

Combinatorics Approach:

The concept behind this proof is to approach it through a combinatorial problem and discover its solution, which indirectly verifies the theorem. Let's explore this straightforward combinatorial problem.

Consider a Necklace chain consisting of beads. There are $$$P$$$ beads in this chain.

References:
Wikepedia