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A clever proof of Euler's Theorem

I've been learning abstract algebra for the last six months or so - and I have learned a lot - yet I get the distinct impression that if Euclid were to appear before me and demand to know why he should care about groups or rings or what have you I wouldn't have much to say. So I was happy to learn of a simple proof of Euler's theorem, which relies solely on Lagrange's theorem, a basic result of group theory.

Euler's theorem deals with ϕ, the Euler totient function, where ϕ(n) is the number of integers less than n that are coprime to n. For example, ϕ(6)=2, since 1 and 5 are the only integers less than 6 that are coprime to 6. If p is prime, ϕ(p)=p1 since every number less than p is relatively prime to p.

Upon first hearing about this function, I thought it was too insanely complex to know anything about. How the hell should I know what ϕ(n) is? There's no easy formula. But it turns out that we can prove the following theorem quite easily:

Euler's theorem: Suppose a,n are coprime. Then aϕ(n)1(modn).

Note that, since ϕ(p)=p1, this is just a generalization of Fermat's little theorem. Here's a three-step proof:

  1. An integer a is invertible modulo n if and only if gcd(a,n)=1 ("invertible" means there's some a1 such that aa11modn). This is the Linear Congruence Theorem.
  2. Consider the group G of all positive integers less than n which are invertible modulo n (formally, G=(Z/nZ)×). ϕ(n) simply counts the number of positive integers a where gcd(a,n)=1, so by (1) there are ϕ(n) elements in G (the order of G is ϕ(n)). You can satisfy yourself that G follows the group axioms quickly.
  3. Lagrange's theorem says that there must be some k such that ak=1 where k divides the order of the group, namely ϕ(n); let's say km=ϕ(n). Then aϕ(n)=akm=(ak)m=1m=1 which proves our theorem!

Isn't that cool? I went from thinking that ϕ was an impossibly complex function to knowing a foundational result about it within 5 minutes, all because of what seemed like an esoteric result in group theory. 90% of this short post is explaining the terms, the proof itself is only a paragraph!

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