show that phi(n)=n/2 iff n=2^k for some positive integer k.
Proving that is a fairly easy so we will prove the converse
I'll present 2 proofs
If is odd, then is not an integer, so is impossible since the LHS is an integer. Thus for that equality to hold, has to be even.
Then write with and
Now since is multiplicative :
If then indeed holds, however if then ( Since b is not coprime to itself now ), thus if we have thus is impossible when and we are done.
We have so we have:
This means that So 2 must be one of the primes there (now the product goes over the prime divisors of n which are greater than 2). But this obviously can't hold ( ) if there's a prime such that . This means that 2 is the only prime dividing and we are done.