## Podcast Beta

## Questions and Answers

What is the value of Ï†(p) for prime p?

If gcd(a, b) = 1, what is the value of Ï†(ab)?

What is the Euler's totient function, Ï†(n), used in?

What is the value of Ï†(p^k) for prime p and k â‰¥ 1?

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What is the definition of Ï†(n), the Euler's totient function?

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How is the greatest common divisor (GCD) calculated in the calculateGCD function?

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If the resulting GCD in the calculateGCD function is equal to 1, x and i are not relatively prime.

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A set of numbers are said to be ____ when their greatest common divisor is 1

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Define what a divisor is according to number theory

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What is the greatest common divisor (GCD) of integers 35 and 21?

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A composite number is a positive integer with no divisors other than itself and 1. (True/False)

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Explain what a reduced residue class mod m is.

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## Study Notes

### Contributions to Number Theory

- Euler's work in number theory is vast and influential, with many concepts and theorems bearing his name.
- He introduced the concept of the Euler's totient function, Ï†(n), which counts the number of positive integers up to n that are relatively prime to n.
- Euler's theorem states that if a and n are coprime, then a^(Ï†(n)) â‰¡ 1 (mod n).

### Euler's Totient Function

- The Euler's totient function, Ï†(n), is a multiplicative function that counts the number of positive integers up to n that are relatively prime to n.
- Ï†(n) is defined as the number of k, 1 â‰¤ k â‰¤ n, such that gcd(k, n) = 1.
- The totient function is used in many areas of number theory, including cryptography, primality testing, and Diophantine equations.
- Euler's totient function has several important properties, including:
- Ï†(p) = p - 1 for prime p
- Ï†(p^k) = p^k - p^(k-1) for prime p and k â‰¥ 1
- Ï†(ab) = Ï†(a)Ï†(b) if gcd(a, b) = 1

### Euler's Contributions to Number Theory

- Euler's work in number theory is vast and influential, with many concepts and theorems bearing his name.

### Euler's Totient Function

- The Euler's totient function, Ï†(n), counts the number of positive integers up to n that are relatively prime to n.
- Ï†(n) is defined as the number of k, 1 â‰¤ k â‰¤ n, such that gcd(k, n) = 1.
- The totient function is used in many areas of number theory, including:
- Cryptography
- Primality testing
- Diophantine equations

- Ï†(n) has several important properties, including:
- Ï†(p) = p - 1, where p is prime
- Ï†(p^k) = p^k - p^(k-1), where p is prime and k â‰¥ 1
- Ï†(ab) = Ï†(a)Ï†(b), where gcd(a, b) = 1

### Euler's Theorem

- Euler's theorem states that if a and n are coprime, then a^(Ï†(n)) â‰¡ 1 (mod n).

### Introduction to Euler's Totient Function

- Number theory is the study of relationships and properties of numbers.
- Euler's totient function, also known as the phi-function, is a fundamental concept in number theory.
- It is denoted by Ï†(n) and represents the number of positive integers less than n that are coprime to n.

### Background

- Divisibility: an integer b is divisible by an integer a if a leaves no remainder when dividing b.
- Greatest Common Divisor (GCD): the largest divisor shared by two or more integers.
- Coprime: a set of numbers are coprime if their GCD is 1.
- Surjective function: a function f: X â†’ Y is surjective if for every element y in Y, there exists an element x in X such that f(x) = y.

### Fundamental Concepts

- Prime numbers: positive integers with no divisors other than themselves and 1.
- Composite numbers: positive integers that are not prime.
- Congruence: an integer a is congruent to an integer b modulo m if a - b is divisible by m.
- Reduced residue class: a set of numbers that are coprime to the modulus m and are not congruent to each other modulo m.

### Patterns of the Totient Function

- Ï†(p) = p - 1 for a prime number p.
- Ï†(pq) = Ï†(p)Ï†(q) for distinct primes p and q.
- Ï†(n) is a multiplicative function: Ï†(mn) = Ï†(m)Ï†(n) for coprime m and n.
- Ï†(pr) = pr - pr-1 for a prime number p.

### The Surjectivity of the Totient Function

- Ï†(x) is never odd for integers x â‰¥ 3.
- Ï†(x) is only odd when x = 2.
- The codomain of the totient function does not contain every even number.
- There exists a lower bound for Ï†(x) where no value can Ï†(x) â‰¥ x2/p.

### Examples and Applications

- Ï†(6) = 2, as the integers less than 6 that are coprime to 6 are 1 and 5.
- Ï†(30) = 8, and we can find 4 distinct x1, x2, x3, x4 such that Ï†(30) = Ï†(x1) = ... = Ï†(x4).
- The totient function can be used to find the number of reduced residue classes modulo n.

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## Description

Explore Euler's influential work in number theory, including his totient function and theorem. Learn about the properties and applications of these fundamental concepts.