Proof of Euler’s Theorem
Let φ(n) = k, and let {a1, . . . , ak} be a reduced residue system mod n.
For some ai in {a1, . . . , ak}
Since (a, n) = 1, {aa1, . . . , aak} is another reduced residue system mod n.
Since this is the same set of numbers mod n as the original system, the two systems must have the same product mod n:
(aa1)· · ·(aak) = a1 · · · ak (mod n)
⇒ ak (a1 · · · ak) = a1 · · · ak (mod n)
Now each ai is invertible mod n, so multiplying both sides by a1−1 · · · ak−1 , We get
ak = 1 (mod n)
or aφ(n) = 1 (mod 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