(a) Show that
$$ \sigma (p^3) = (p^2 + 1) (p + 1) $$
where $p$ is prime.
(b) Show that
$$ \sigma (p^5) = (p^2 − p + 1) (p^2 + p + 1) (p + 1) $$
where $p$ is prime.
(a) By definition of the sigma function
$$ \sigma (p^3) = 1 + p + p^2 + p^3 = (p^2+1)(p+1) $$
(b) By definition of the sigma function
$$ \sigma(p^5) = 1 + p + p^2 + p^3 + p^4 + p^5 = (p + 1) (p^2 - p + 1) (p^2 + p + 1) $$