Illustrate the proof of Wilson's theorem for p=17 by pairing the integers 2,3,4, …, 15 and using

Illustrate the proof of Wilson's theorem for p=17 by pairing...

Key Concepts

Multiplicative Inverses

In modular arithmetic, particularly modulo a prime number, every nonzero element has a unique multiplicative inverse. This concept is essential when pairing elements in a product because each number can be paired with its inverse, so that their product is congruent to 1 modulo the prime, simplifying the overall product computation in proofs such as that of Wilson's Theorem.

Pairing Technique

The pairing technique involves grouping numbers with their multiplicative inverses in the calculation of a factorial modulo a prime. By pairing each number with its unique inverse (except self-inverses), one reduces the complexity of the product, as the product of each pair is congruent to 1 modulo the prime. This method is a powerful tool in simplifying the verification of Wilson's Theorem and related modular calculations.

Wilson's Theorem

Wilson's Theorem is a fundamental result in number theory which states that for any prime number p, the factorial of (p-1) is congruent to -1 modulo p; that is, (p-1)! ? -1 (mod p). This theorem provides a useful criterion for testing the primality of a number and has deep connections with the structure of the multiplicative group modulo p.

Modular Arithmetic

Modular arithmetic involves performing arithmetic operations where numbers 'wrap around' after reaching a certain modulus. In the context of Wilson's Theorem, it is used to simplify large factorial expressions by considering them under modulo p, which often reduces computations significantly and reveals cyclic patterns in the multiplicative group of integers modulo a prime.

Transcript

00:01 In this question, we are actually required to prove the famous little theorem, which can be stated as, suppose we have a prime number p and we have an integer a.

00:17 If p does not divide a, then we know a to the power p is equal to a mod p.

00:25 Okay, let's follow the hints.

00:27 We want to finish this proof step by step.

00:31 For the first part, we are required to show that 1 times a, 2 times a, 3 times a, 2, p minus 1 times a.

00:48 For those p minus 1 numbers we want to show, there are two we are required to show.

00:59 No two of them are congruent mod p.

01:04 That means we want to show for any i and j, which is greater or equal to 1 and less or equal to p minus 1.

01:18 If i is not equal to j, we say i times a is not equal to j times a mod p.

01:28 Okay, this is what we want to prove.

01:34 We want to prove it by the contradiction, assume not.

01:42 Then we know there exists i is not equal to j such that i times a is just equal to j times a mod p.

01:56 Okay, this is equivalent to say i times a minus j times a can be divided by p.

02:12 I mean p just divides this number.

02:16 Okay, this is just equal to i minus j times a.

02:21 Now notice p is a prime number.

02:26 That means once p divides a product of two things, then we know p divides i minus j or p can divide a.

02:41 However, by our assumption, this cannot happen.

02:45 That means p must divide i minus j.

02:48 But notice i minus j is some number less than p.

02:59 I mean the difference between i and j is some number strictly less than p.

03:05 That means it is impossible either.

03:10 That means our assumption is not right.

03:15 So for any i and j from 1 to p minus 1, if they are different, then we know they can't be equal.

03:28 With respect to modulo p.

03:31 Okay, now we just prove the first part of our question.

03:35 For the second part, we are required to show p times 1 times 2 times 3 times p minus 1 is equal to a times 2 to a times 3a times p minus 1 times a modulo.

04:07 Okay, to prove this identity, we just need to consider all of those numbers.

04:18 1, 2, 3, 2p minus 1 and 2a, 3a, and all of those numbers in this group, the group formed by modulo p.

04:31 Okay, in this group, we know 1, 2, 3, 2p minus 1, all of them just represent 0.

04:49 Maybe we can include 0.

04:52 Those n numbers just represent n different equivalent classes.

05:04 Okay, now let's consider the other classes.

05:12 I mean a, 2a, 3a, 2p minus 1 times a.

05:21 Okay, i mean we just regard all of those as an element in zp.

05:27 That means a, 2a, 3a, and p minus 1 times a must be contained in some equivalent classes represented by the number here...

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