Fermat Theorem Combinatorics Proof

Revision en21, by ASHWANTH_K, 2024-08-13 22:19:00

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

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en42 English ASHWANTH_K 2024-08-14 07:15:35 6 Tiny change: ' \n\nThus, this comple' -> ' \n\nThis comple'
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en40 English ASHWANTH_K 2024-08-14 07:10:21 47 Tiny change: 'ons. \n\n<spoiler' -> 'ons. \n![ ](https://ibb.co/DrMDhmJ)\n<spoiler'
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en27 English ASHWANTH_K 2024-08-13 22:41:17 4 Tiny change: 'consider $A = 2, P = 3$ ' -> 'consider $a = 2, p = 3$ '
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en22 English ASHWANTH_K 2024-08-13 22:28:39 416 Tiny change: ' $ = 2^4$ \n $ = 16$ ' -> ' $ = 2^4 = 16$ '
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en12 English ASHWANTH_K 2024-08-13 22:09:27 2 Tiny change: 'v 1 \pmod{n}$ \n\n' -> 'v 1 \pmod{p}$ \n\n'
en11 English ASHWANTH_K 2024-08-13 22:09:16 11 Tiny change: ' \equiv 1 mod p$ \n\n\' -> ' \equiv 1 \pmod{n}$ \n\n\'
en10 English ASHWANTH_K 2024-08-13 22:08:23 8 Tiny change: ' $a^{p-1} = 1 mod ' -> ' $a^{p-1} \equiv 1 mod '
en9 English ASHWANTH_K 2024-08-13 22:07:51 4 Tiny change: '{p-1} = 1 mod p$ \n\' -> '{p-1} = 1 mod p$ \n\'
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en1 English ASHWANTH_K 2024-08-13 22:00:23 307 Initial revision (saved to drafts)