Proving n/Φ(n)=2q/q-1: A Proof Using Euler's Totient Function

[AlexHall]

Hello, can anyone help with this question? Thank you.


Let n even perfect number and q prime. Show that n/Φ(n)=2q/q-1.

Φ(n) is the Euler function-totient (the number of positive integers less than or equal to n that are coprime to n)


I have tried euler-euclid theorem but could not get it.

 

[Office_Shredder]

2q/(q-1) is different for different primes q, so you must have some additional condition on what q is (unless you really meant without the parentheses, in which case 2q/q-1 = 1 which makes no sense). Unless you're trying to show such a q exists?

 

[AlexHall]

phi(n)=phi[2^(k-1)q] where q=(2^k)-1

phi(n)=phi(2^(k-1))phi(q)=phi(2^(k-1))(q-1)

Is there any property I can use to finish this?

 

[Dick]

What's phi(2^(k-1))? There's a formula for phi for powers of a prime. (It's also the number of odd integers less than 2^(k-1)).