Euler’s Theorem
What is Euler’s Theorem?
Euler’s Theorem verifies that if a and n are coprime and positive integers, then aϕ(n) ≡ 1 (mod n), where ϕ(n) represents the result of Euler’s totient function, i.e. the number of positive integers less than n that are coprime to n.
What is the significance of Euler’s Theorem?
Euler’s Theorem holds importance to number theory and cryptology. That result forms the base for all the other outcomes that provide an understanding for the properties of modular arithmetic and is necessary for various cryptographic systems like RSA encryption system.
Can Euler’s Theorem be applied to composite moduli?
The fact that the Euler’s Theorem can be applied to composite moduli becomes clear, provided that for the condition a and n are coprime positive integers to be satisfied.
How can we use Euler’s Theorem even today?
Euler’s Theorem has been of relevance in number theory and its extension in cryptography with the main application to RSA encryption. Besides, it is also a tool in the number theory, group theory
What are Euler’s Theorem’s general expressions?
Euler’s Theorem, sometimes extended to Carmichael’s Theorem, Fermat’s Little Theorem, and multiple integers, forms the foundations of number theory.
Euler’s Theorem
Euler’s Theorem states that for any integer a that is coprime with a positive integer m, the remainder of aϕ(m) divided by m is 1. We focus on proving Euler’s Theorem because Fermat’s Theorem is essentially a specific instance of it. This relationship arises because when p is a prime number, ϕ(p) equals p-1, thus making Fermat’s Theorem a subset of Euler’s Theorem under these conditions.
Euler’s theorem is a fundamental result in number theory, named after the Swiss mathematician Leonhard Euler. It states a relationship between the number theory functions and concepts of modular arithmetic. In this article, we will discuss Euler’s Theorem, including its statement and proof.
Table of Content
- What is Euler’s Theorem?
- Euler’s Theorem Formula
- Historical Background of Euler’s Theorem
- Proof of Euler’s Theorem
- Applications of Euler’s Theorem
- Euler’s Theorem Examples
- Practice Questions on Euler’s Theorem