Refer to the definition in Exercise 12 of Euler's phi function. (a) Compute ϕ(7), ϕ(20), and ϕ(1

Refer to the definition in Exercise 12 of Euler's phi...

Key Concepts

Euler's Totient Function

Euler's Totient Function, often denoted as ?(n), is a fundamental arithmetic function that counts the positive integers up to n that are coprime to n. It plays a crucial role in number theory and particularly in understanding the structure of integers under multiplication modulo n. The computation of ?(n) relies on the prime factorization of n, using the formula ?(n) = n * ?(1 - 1/p) for each distinct prime divisor p of n.

Euler's Theorem

Euler's Theorem states that if n is a positive integer and a is an integer coprime to n, then a^(?(n)) is congruent to 1 modulo n. This theorem is pivotal in reducing large exponents in modular arithmetic calculations, thereby simplifying computations involving exponentiation modulo n. It serves as the theoretical foundation for many efficient algorithms in number theory.

Modular Arithmetic

Modular arithmetic deals with integers and their remainders upon division by a fixed positive integer, known as the modulus. It forms the basis for working with congruences, where two numbers are considered equivalent if they leave the same remainder when divided by the modulus. This framework is essential for solving problems in number theory that involve cyclic structures and residue classes.

Efficient Modular Exponentiation

Efficient modular exponentiation involves methods for computing large powers modulo a number without directly calculating the potentially huge exponentiation result. Techniques such as exponentiation by squaring and application of Euler's Theorem allow one to break down and simplify the computation, making it feasible to compute expressions like a^b (mod n) even when b is large.

Transcript

00:01 All right, so here we're looking at oil or functions, and the oiler function of a number, such as an a, which is 4, is going to be equal to the number of positive integers less than are equal to that number, relatively prime.

00:20 And so what that would mean is we're looking at the number of integers from 1, 2, 3, and 4 that are relatively prime to 4.

00:32 So we can start with finding the greatest common divisor, right? or we can use prime factorization.

00:46 So the greatest common divisor of 4 and 1 is simply going to be 1.

00:55 Right? we can use the greatest common divisor of 4 and 2 is equal to 2.

01:05 Greatest common divisor of 4 and 3 is equal to 1.

01:10 And the greatest common divisor of four and four is going to be equal to four.

01:20 Okay, so we had two values where the number was one, greatest common divisor is one.

01:27 So that means there were two of these four are relatively prime.

01:32 And so that would mean that the oil or function of four is equal to two, just the number of relatively prime numbers.

01:42 Okay, so we can go to b, and b is asking us what the oiler motion of 10 is.

01:53 And so this time i'm going to show us how to do it using prime factors, right? so i just want the prime factor of everything 1 through 10.

02:04 So 1 is just going to be 1, 2 is prime, that's 2, 3 is prime, that's 3.

02:10 4 is going to be 2 squared.

02:13 5 is also just prime.

02:15 6 would be 2 times 3, 7 ,000.

02:18 Is itself.

02:20 Eight is two cubed.

02:23 Nine is three times three and then ten is two times five...

Please give Ace some feedback