I am trying to decipher the proofs to the following statements: Let $d$ be the divisor function, then;
- show that $d(n)$ is odd if and only if $n$ is a square
- Show that for a given $n\geq 2$, there are infinitely many positive integers $x$ such that $d(x) = n$.
I am trying to go through them, but I keep getting stuck along the way. Help answering the question would be appreciated
Hint for the first : If $n=p_1^{a_1}...p_k^{a_k}$, then $d(n)=(a_1+1)...(a_k+1)$
Note that all factors have to be odd, if $d(n)$ is odd.
Hint for the second : Use $N=p^{n-1}$ , $p$ any prime.