91Ó°ÊÓ

Euler's totient function \( \phi(n) \) is defined as the count of numbers in {1, 2, 3, ..., \( n \)} that are relatively prime to \( n \). The properties of \( \phi(n) \) are important to consider. One property is that for a prime number \( p \), \( \phi(p) = p-1 \). Another notable property is that if \( n \) is the product of two prime numbers \( p \) and \( q \), then \( \phi(n) = (p - 1)(q - 1) \).

Firstly, consider prime \( n \). For prime \( n = 2 \), \( \phi(2) = 2-1 = 1 = 2^0 \). For prime \( n \neq 2 \), \( \phi(n) = n - 1 \) which is an odd number and couldn't be a power of \( 2 \). Hence, the only prime number \( n \) such that \( \phi(n) \) is a power of \( 2 \) is \( 2 \).

Next, consider composite \( n \). If \( n = p \cdot q \) is the product of two distinct primes, then \( \phi(n) = (p - 1)(q - 1) \), which isn't a power of two unless \( p = 2 \) or \( q = 2 \). But in both cases, \( \phi(n) \) isn't a power of \( 2 \) because for \( p = 2 \), \( \phi(n) = q - 1 \) which is odd, and for \( q = 2 \), \( \phi(n) = p - 1 \), which is odd too (remember that the only even prime number is \( 2 \)). If \( n = p^2 \), for some prime \( p \), then \( \phi(n) = p^{2-1}(p - 1) \) which isn't a power of \( 2 \) except when \( p = 2 \). So then, \( n = 2^2 = 4 \) is such that \( \phi(n) = 2 \), which is a power of \( 2 \). If \( n = p^k \), for \( k > 2 \), then \( \phi(n) = p^{k-1}(p - 1) \) which isn't a power of \( 2 \). Thus, the only composite \( n \) that makes \( \phi(n) \) turn out to be a power of \( 2 \) is \( n = 4 \).

After carefully analyzing pairs of co-primes \( n \) and \( \phi(n) \), both for prime and composite integers, the only cases in which \( \phi(n) \) is a power of \( 2 \) are \( n = 2 \) and \( n = 4 \).