Question 7
A positive integer is called perfect if it equals the sum of its positive divisors other than itself.
a) Prove that an odd perfect number has the form $n=p^s m^2$, where $p$ is a prime of the form $4k+1$, $s$ is a positive integer of the form $4h+1$, and $m$ is a positive integer not divisible by $p$.
b) Find all integers $n\gt 1$ for which both $n-1$ and $n(n+1)/2$ are perfect.
a) Prove that an odd perfect number has the form $n=p^s m^2$, where $p$ is a prime of the form $4k+1$, $s$ is a positive integer of the form $4h+1$, and $m$ is a positive integer not divisible by $p$.
b) Find all integers $n\gt 1$ for which both $n-1$ and $n(n+1)/2$ are perfect.