What is Euler’s Theorem?
Euler’s Theorem is a fundamental concept in number theory. It states that if n and a are coprime positive integers, meaning that they have no mutual proper dividers other than m=1, then aϕ(n) and 1 are relative primes in modulo n.
Euler’s Theorem is a generalization of Fermat’s Little Theorem and serves as a basis for simplifying complex problems into computationally less expensive ones.
Euler’s Theorem Statement
Euler’s Theorem states if a and n are coprime positive integers, then:
aϕ(n) ≡ 1 (mod n)
Where,
- ϕ(n) is Euler’s totient function, and
- ≡ denotes equivalence,
- mod n represents congruence modulo n.
Euler’s Totient Function
Formally, for a positive integer n, ϕ(n) is defined as follows:
ϕ(n) = count of integers 1 ≤ a <n such that gcd (a,n)=1
Where:
- gcd(a,n) denotes the greatest common divisor of a and n.
- ϕ(n) represents the totient function of n.
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