- #1
cragar
- 2,552
- 3
Homework Statement
Prove: [itex] \phi (n) [/itex] is a power of 2 if and only if
[itex] n=(2^x)p_1...p_k [/itex]
where the [itex] p_i [/itex] are distinct Fermat primes. And it doesn't say but I am pretty sure x
is a positive integer. and [itex] \phi [/itex] is Eulers phi function.
The Attempt at a Solution
we know that phi is a multiplicative function when the gcd(q,w)=1
so [itex] \phi (n)= \phi (2^x) \phi (p_1)...\phi (p_k) [/itex]
and [itex] \phi (2^x) = 2^x-2^{x-1}=2^{x-1}(2-1) [/itex] so that is a power of 2.
And each Fermat prime will be [itex] \phi (p_i)=2^{2^s}+1-1 [/itex] so each Fermat prime will be a power of 2 when phi is applied to it. So I guess I have showed that if n is written as it is then
phi (n) will be a power of 2, but I haven't shown the other direction that this is the only n where phi (n) is a power of 2. Maybe I could try a proof by contradiction.