Phi Function Question?

Question

What can one say about m when the value of φ(m) is a prime number? Also, what can one say about m when φ(m) is the square of a prime number?

Answer 1

If phi(m) is prime, then m=3, m=4, or m=6.

That's because, for prime p, phi(p^n) = p^(n-1)(p-1)

So phi(p^n) is not prime for n>2, or for n=2 and p>2.
For n=2, p=2, phi(4) = 2, so m=4 is one solution.

If n=1, then phi(p^n) = phi(p) = p-1. This can't be prime for p>3 because p-1 is even and bigger than 2. So we only need to try phi(2) and phi(3). m=3 is the only solution here.

In general, if m is not a prime power, then factoring m into prime powers, we get:

m = p1^a1 . p2^a2 ...

phi(m) = phi(p1^a1) . phi(p2^a2) . ..

phi(m) can only be prime if one of the phi(pi^ai) is prime and the rest are 1. The only integer for which phi(n)=1 is n=2.
We can see then that the only solution is m=2*3=6.

There are similarly very few m's such that phi(m) is a prime squared. m=8, m=5, m=10, m=12. Basically, the (p-1) factor of phi(p^n) makes phi even for p>2, so phi(m) would have to be 2^2 for m having an odd prime factor.

Answer 2

Note that φ(m) is always even. So if it
is a prime number, it must equal 2.
The only values of m for which this happens
are m = 3, 4 and 6.
Similarly, if φ(m) is the square of a prime number
it must equal 4.
The only values of m for which this happens
are m = 5, 8, 10 and 12.