Exercise 24, § 1

a) Let $n = \prod_k p_k^{v_k} > 0$ be a rational integer expressed in terms of its prime factors. Show that the sum of the positive divisors of n is given by the formula

b) We say that n is perfect if $2n = \sigma(n)$. Show that an even number n is perfect if and only if it has the form $2^h(2^{h+1}-1)$, where $h \geq 1$ and $2^{h+1}-1$ is prime. (If n has the form $2^m q$, where $m \geq 1$ and q is odd, show that q is divisible by $2^{m+1}-1$, and deduce that, if q were not prime, then we would have $\sigma(n) > 2n$).