If p is a prime number, prove that for any integer a, a^p=a...

Key Concepts

Prime Numbers

Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves. They form the building blocks of number theory and are essential in understanding various properties of integers, especially in problems involving divisibility and modular arithmetic.

Modular Arithmetic

Modular arithmetic deals with the arithmetic of remainders when integers are divided by a given modulus. It provides a framework for understanding congruences and is widely used in solving problems related to divisibility, cyclic groups, and residue classes. In this context, it is used to show that two numbers are equivalent with respect to a modulus.

Fermat's Little Theorem

Fermat's Little Theorem is a fundamental result in number theory stating that if p is a prime number and a is any integer, then a raised to the power of p is congruent to a modulo p. This theorem is significant because it reveals a deep relationship between prime numbers and exponentiation in modular arithmetic, and it is employed in various proof techniques and algorithms, including those in cryptography.

Binomial Theorem

The Binomial Theorem provides a method to expand expressions raised to a power as a sum of terms involving binomial coefficients. In many proofs of Fermat's Little Theorem, the Binomial Theorem is used to expand (1 + k)^p and analyze the divisibility properties of the resulting binomial coefficients, particularly taking advantage of the fact that these coefficients are divisible by p when p is prime.

Transcript

00:01 Okay, here we are given some prime number p.

00:04 We want to prove a to the power p is just equal to a mod p for any integer a.

00:13 Okay, this is a very famous theory.

00:17 To begin with our proof, we just notice if a is some positive integer and union.

00:31 Let's just consider the positive integer.

00:36 For a positive if, if we have this, if this is true, then we know now minus a must be an active integer.

00:59 Then as the mod relation is closed under multiplication, that means if we have this, we can multiply minus one on both sides.

01:16 We have minus one times a to the power p is equal to minus one times a mod p.

01:27 Okay, now as p is a prime number, so it must be odd.

01:32 That means this term is actually equal to minus a to the power p is equal to minus a mod p.

01:43 So we can see if this relationship is true for any positive integer, it must be true for an active integer.

01:58 Now let's just, so we only need to consider a greater or equal to zero and a is an integer.

02:07 Because once we can prove that, we just, we only need to use the minus one, the multiplication, we only need to multiply a minus one to make all, to make the relationship, to make this relation true for all integers.

02:26 Okay, first notice zero to the power p is equal to zero mod p.

02:39 This is trivial, right? and one to the power p is equal to one.

02:53 Observe, by this observation, we know we want to prove it by induction.

03:04 Suppose for any s greater or equal to zero and less or equal to k, we have s to the p is equal to s mod p.

03:18 Let's consider k plus one.

03:22 Once we can prove it is also true for k plus one, we know by the inductive hypothesis, this must be true for every natural number.

03:32 And as i said, once we prove this statement for a natural number, we can prove it is also true for any negative integers.

03:45 And combine all of those facts, we can show it is true for all integers.

03:50 So the key point is to prove the situation for k plus one and k plus one to the power p by the so -called binomial formula.

04:10 It can be written as now we have k n goes from zero to p.

04:17 Here we have p choose n, and k to the power n times one to the power p minus n.

04:29 Okay, simplify it.

04:30 We have goes from zero to p, this is our coefficient, and one to some power...

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