Articles containing proofs | Algebra | Multiplicative functions | Modular arithmetic | Number theory
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n. For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1. Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n).This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring ). It is also used for defining the RSA encryption system. (Wikipedia).
Introduction to Euler's Totient Function!
Euler's totient function φ(n) is an important function in number theory. Here we go over the basics of the definition of the totient function as well as the value for prime numbers and powers of prime numbers! Modular Arithmetic playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJ
From playlist Modular Arithmetic
Number Theory | Euler's Totient Function: Definition and Basic Example
We define Euler's totient function and give some basic examples. http://www.michael-penn.net
From playlist Mathematics named after Leonhard Euler
Number Theory | A Formula for Euler's Totient Function
We present a formula for Euler's totient function. http://www.michael-penn.net
From playlist Mathematics named after Leonhard Euler
Explicit Formula for Euler's Totient Function!
Totient of p^a: https://youtu.be/NgZ33qr5WHM?t=210 Product formula: https://youtu.be/qpYqvNBQZ4g Euler's totient function involves counting how many numbers are coprime to n. In fact, we can calculate this value directly as long as we know the prime factors! This makes many theorems in n
From playlist Modular Arithmetic
Reciprocals, powers of 10, and Euler's totient function II | Data Structures Math Foundations 203
We introduce the idea of the unit group U(n) of a natural number n. This is an algebraic object that contains important data about how multiplication mod n works, even for a composite number n. There is a natural connection with Euler's totient function, and we will see how to exploit this
From playlist Math Foundations
Eulers Theorem - Applied Cryptography
This video is part of an online course, Applied Cryptography. Check out the course here: https://www.udacity.com/course/cs387.
From playlist Applied Cryptography
Number Theory | The Multiplicativity of Euler's Totient Function
We state and prove when Euler's totient function is multiplicative. http://www.michael-penn.net
From playlist Number Theory
Introduction to number theory lecture 14. Euler's totient function
This lecture is part of my Berkeley math 115 course "Introduction to number theory" For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8 We cover the basic properties of Euler's totient function. The textbook is "An introducti
From playlist Introduction to number theory (Berkeley Math 115)
Discrete Structures: Multiplicative Inverse; Greatest Common Divisor; Euler's Totient Function
Decrypting the linear cipher leaves us with a fundamental problem: dividing two integers yields a fraction, which is difficult to work with. Learn about new concepts: the greatest common divisor (GCD), the multiplicative inverse, and Euler's totient function. These will allow us to decrypt
From playlist Discrete Structures, Spring 2022
Euler's Totient Theorem and Fermat's Little Theorem - Complete Proof & Intuition
Video on coprime numbers mod n: https://youtu.be/SslPWR2N5jA Video on the cancellation rule for modular arithmetic: https://youtu.be/UvnVghpIjwk Euler's theorem relates to modular exponentiation. Fermat's little theorem is a special case for prime modulus. Here we go through an explanatio
From playlist Modular Arithmetic
Discrete Structures: Multiplicative inverse, Euler's totient function, and Euler's theorem
This is a continuation of the previous live stream session. Learn more about Euler's totient function and how we can use it, along with Euler's theorem, to compute the multiplicative inverse of any number (a mod n). We'll also learn about the extended Euclidean algorithm to compute the mul
From playlist Discrete Structures, Spring 2022
GT12. Aut(Z/n) and Fermat's Little Theorem
Abstract Algebra: We show that Aut(Z/n) is isomorphic to (Z/n)*, the group of units in Z/n. In turn, we show that the units consist of all m in Z/n with gcd(m,n)=1. Using (Z/n)*, we define the Euler totient function and state and prove Fermat's Little Theorem: if p is a prime, then, for
From playlist Abstract Algebra
AKPotW: A Lack of Primitive Roots [Number Theory]
If this video is confusing, be sure to check out our blog for the full solution transcript! https://centerofmathematics.blogspot.com/2018/05/advanced-knowledge-problem-of-week-5-3.html
From playlist Center of Math: Problems of the Week
I talk about the basics of the Key sharing algorithm in cryptography.
From playlist Cryptography
Proof that the Totient Function is Multiplicative
Coprime numbers mod n: https://youtu.be/SslPWR2N5jA Chinese remainder theorem: https://www.youtube.com/playlist?list=PL22w63XsKjqyg3TEfDGsWoMQgWMUMjYhl Surjection and bijection: https://youtu.be/kt5eABzTVGQ Explanation of why Euler's totient function of a product of coprime numbers is e
From playlist Modular Arithmetic
Number Theory | Euler's Totient Function and Powers of Primes
We give a formula for the value of Euler's totient function on powers of primes. http://www.michael-penn.net
From playlist Number Theory
CTNT 2020 - The global field Euler function. Santiago Arango-Piñeros.
The paper is available at https://arxiv.org/abs/2005.04521?fbclid=IwAR34njBRG6gEAjzQqdk7johkPEC5i4c5Bbq1MJtyeNAZ95yeQWvaiys2LF0 Comments very welcome!
From playlist CTNT 2020 - Conference Videos