examday

/TMUA

Consider the following theorem:

If $2^k+1$ is a prime, then $k$ is a power of $2$. (*)

Which of the following statements, taken individually, is/are equivalent to (*)?

I If $k$ is a power of $2$, then $2^k+1$ is prime.

II $2^k+1$ is not prime only if $k$ is not a power of $2$.

III A sufficient condition for $k$ to be a power of $2$ is that $2^k+1$ is prime.