Fermat’s and Euler’s Theorem
Euler’s Theorem is generalization of fermat’s theorem. Here are the key differences between Fermat’s and Euler’s Theorem:
Feature | Fermat’s Little Theorem | Euler’s Theorem |
---|---|---|
Conditions | Applicable when p is prime | Applicable for any positive integer n and a relatively prime to n |
Statement | If p is prime and a is not divisible by p, then ap−1 ≡ 1 (mod p) | If n is a positive integer and a is relatively prime to n, then aϕ(n) ≡ 1 (mod n) |
Converse | The converse is not always true. | The converse is true: If aϕ(n) ≡ 1 (mod n), then a and n are coprime. |
Requirement | Requires the modulus p to be prime | Does not require the modulus to be prime |
Generalization | Less general, restricted to prime moduli | More general, applicable to any positive integer modulus |
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
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