Articles containing proofs | Modular arithmetic | Theorems in number theory
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, and is Euler's totient function, then a raised to the power is congruent to 1 modulo n; that is In 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is not prime. The converse of Euler's theorem is also true: if the above congruence is true, then and must be coprime. The theorem is further generalized by Carmichael's theorem. The theorem may be used to easily reduce large powers modulo . For example, consider finding the ones place decimal digit of , i.e. . The integers 7 and 10 are coprime, and . So Euler's theorem yields , and we get . In general, when reducing a power of modulo (where and are coprime), one needs to work modulo in the exponent of : if , then . Euler's theorem underlies the RSA cryptosystem, which is widely used in Internet communications. In this cryptosystem, Euler's theorem is used with n being a product of two large prime numbers, and the security of the system is based on the difficulty of factoring such an integer. (Wikipedia).
This video given Euler's identity, reviews how to derive Euler's formula using known power series, and then verifies Euler's identity with Euler's formula http://mathispower4u.com
From playlist Mathematics General Interest
How to derive Euler's formula using differential equations! Free ebook http://bookboon.com/en/introduction-to-complex-numbers-ebook A somewhat new proof for the famous formula of Euler. Here is the famous formula named after the mathematician Euler. It relates the exponential with cosin
From playlist Intro to Complex Numbers
Euler's formulas, Rodrigues' formula
In this video I proof various generalizations of Euler's formula, including Rodrigues' formula and explain their 3 dimensional readings. Here's the text used in this video: https://gist.github.com/Nikolaj-K/eaaa80861d902a0bbdd7827036c48af5
From playlist Algebra
Theory of numbers: Congruences: Euler's theorem
This lecture is part of an online undergraduate course on the theory of numbers. We prove Euler's theorem, a generalization of Fermat's theorem to non-prime moduli, by using Lagrange's theorem and group theory. As an application of Fermat's theorem we show there are infinitely many prim
From playlist Theory of numbers
Differential Equations | The solution of a Cauchy-Euler Differential Equation
We prove a general theorem regarding the form of a solution of a Cauchy-Euler Differential Equation. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Mathematics named after Leonhard Euler
Number Theory | Euler's Theorem Example 1
We present an example problem that uses Euler's theorem. http://www.michael-penn.net
From playlist Number Theory
Differential Equations | Euler's Method
We derive Euler's method for approximating solutions to first order differential equations.
From playlist Mathematics named after Leonhard Euler
Proving Euler's Formula (2 of 4: Differentiating both sides)
More resources available at www.misterwootube.com
From playlist Introduction to Complex Numbers
MA 15: Euler circuits and paths
This video is for my Spring 2020 section of MA 15, for the class meeting on Friday April 3. Fast forward music is from "Now Get Busy" by the Beastie Boys, licensed Creative Commons Noncommercial Sampling Plus.
From playlist Math 15 Spring 2020
Number Theory | Euler's Theorem Proof
We present a proof of Euler's Theorem. http://www.michael-penn.net
From playlist Mathematics named after Leonhard Euler
Four theorems about the Euler characteristic and some space invaders
A talk about Euler characteristics and digital topology meant for a general quantitatively literate audience- hopefully understandable to anybody who can handle basic mathematical ideas. I gave this talk at the weekly colloquium for the Fairfield University summer research groups, includin
From playlist Research & conference talks
Math Explorations Ep22, Euler circuits & paths (Mar 23, 2022)
This is a recording of a live class for Math 1015, Mathematics: An Exploration, an undergraduate course for non-technical majors at Fairfield University, Spring 2022. The major topics are voting, gerrymandering, and graph theory. Handouts and homework are at the class website. Class web
From playlist Math 1015 (Mathematical Explorations) Spring 2022
Andrew Thomas (7/1/2020): Functional limit theorems for Euler characteristic processes
Title: Functional limit theorems for Euler characteristic processes Abstract: In this talk we will present functional limit theorems for an Euler Characteristic process–the Euler Characteristics of a filtration of Vietoris-Rips complexes. Under this setup, the points underlying the simpli
From playlist AATRN 2020
The second most beautiful equation and its surprising applications
Get free access to over 2500 documentaries on CuriosityStream: https://curiositystream.com/majorprep (use promo code "majorprep" at sign up) STEMerch Store: https://stemerch.com/ Support the Channel: https://www.patreon.com/zachstar PayPal(one time donation): https://www.paypal.me/ZachStar
From playlist Applied Math
Dipendra Prasad - Branching laws: homological aspects
By this time in the summer school, the audience will have seen the question about decomposing a representation of a group when restricted to a subgroup which is referred to as the branching law. In this lecture, we focus attention on homological aspects of the branching law. The lecture
From playlist 2022 Summer School on the Langlands program
Math for Liberal Studies - Lecture 1.2.2 Practice with Euler's Theorem
This is the second video lecture for Math for Liberal Studies Section 1.2: Finding Euler Circuits. In this video, we work through several examples of applying Euler's Theorem to determine whether or not a graph has an Euler circuit.
From playlist Math for Liberal Studies Lectures
Ext-analogues of Branching laws – Dipendra Prasad – ICM2018
Lie Theory and Generalizations Invited Lecture 7.5 Ext-analogues of Branching laws Dipendra Prasad Abstract: We consider the Ext-analogues of branching laws for representations of a group to its subgroups in the context of p-adic groups. ICM 2018 – International Congress of Mathematic
From playlist Lie Theory and Generalizations
Graph Theory: 24. Euler Trail iff 0 or 2 Vertices of Odd Degree
I begin by reviewing the proof that a graph has an Euler tour if and only if every vertex has even degree. Then I show a proof that a graph has an Euler trail if and only it has either 0 or 2 vertices of odd degree. An introduction to Graph Theory by Dr. Sarada Herke. Related Videos: htt
From playlist Graph Theory part-5
Euler’s method - How to use it?
► My Differential Equations course: https://www.kristakingmath.com/differential-equations-course Euler’s method is a numerical method that you can use to approximate the solution to an initial value problem with a differential equation that can’t be solved using a more traditional method,
From playlist Differential Equations