derivation of Euler phi-function
In this “proof” we will construct the solution for the Euler phi-function, ϕ(n)=n∏p|n(1-1n).
We will do this for the natural number n>0.
Keep in mind that gcd(a,n)=1⟺ a is not divisible by p for all primes p dividing n.
Let n≥2 and p1,p2,⋯,pr be all prime divisors of n.
Let N={a∣0≤a<n,gcd(a,n)=1} and Ai:=.
If than .
Thus,
Using inclusion-exclusion,
Title | derivation of Euler phi-function |
---|---|
Canonical name | DerivationOfEulerPhifunction |
Date of creation | 2013-03-22 17:42:52 |
Last modified on | 2013-03-22 17:42:52 |
Owner | jwaixs (18148) |
Last modified by | jwaixs (18148) |
Numerical id | 22 |
Author | jwaixs (18148) |
Entry type | Derivation |
Classification | msc 11-00 |