91Ó°ÊÓ

If \(n\) is a perfect number, prove that \(\sum_{d i n} 1 / d=2\).

(a) If \(n>6\) is an even perfect number, prove that \(n \equiv 4(\bmod 6)\). [Hint: \(2^{p-1} \equiv 1(\bmod 3)\) for an odd prime \(p\).] (b) Prove that if \(n \neq 28\) is an even perfect number, then \(n \equiv 1\) or \(-1(\bmod 7)\).

Establish that any Fermat prime \(F_{n}\) can be written as the difference of two squares, but not of two cubes. | Hinr: $$ \left.F_{n}=2^{2^{2}}+1=\left(2^{2^{2}-1}+1\right)^{2}-\left(2^{2^{2}-1}\right)^{2} \cdot\right] $$