Fermat Theorem Combinatorics Proof

Revision en23, by ASHWANTH_K, 2024-08-13 22:30:11

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.

Problem

Consider a Necklace chain consisting of beads. There are $$$p$$$ beads in this chain. You are allowed to color each bead with $$$a$$$ possible colors available. The colors are available in infinite amounts, there is no other restriction on coloring. Find the number of ways to color these beads.

It can be easily shown that there are $$$a^p$$$ ways to color the beads to get different necklaces. (considering cyclic rotations as different)

References:
Wikepedia

History

 
 
 
 
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en42 English ASHWANTH_K 2024-08-14 07:15:35 6 Tiny change: ' \n\nThus, this comple' -> ' \n\nThis comple'
en41 English ASHWANTH_K 2024-08-14 07:15:05 110 Tiny change: 'ime and $a$ is any i' -> 'ime and $a > 1$ is any i'
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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en38 English ASHWANTH_K 2024-08-14 00:51:19 0 Tiny change: ' characters are repea' -> ' character are repea' (published)
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en35 English ASHWANTH_K 2024-08-14 00:40:07 33 Tiny change: ' \n\nReferenc' -> ' \n\nThus, completing the proof. \nReferenc'
en34 English ASHWANTH_K 2024-08-14 00:39:37 1091 Tiny change: 'xplanation and Proof">\nLet's ' -> 'xplanation">\nLet's '
en33 English ASHWANTH_K 2024-08-14 00:20:03 1135 Tiny change: 'inal form.`\n######Pr' -> 'inal form.\n######Pr'
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en28 English ASHWANTH_K 2024-08-13 22:43:22 139
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$ '
en21 English ASHWANTH_K 2024-08-13 22:19:00 8 Tiny change: 'mple:\n Say a =' -> 'mple:\n Say a ='
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en18 English ASHWANTH_K 2024-08-13 22:16:33 13 Tiny change: 'e Theorem:\n- States that' -> 'e Theorem: \n It states that'
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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)