?

Number Theory

Free Version

Upgrade subject to access all content

Moderate

Perfect Number is Related to Mersenne Prime

NUMTH-HFJRAW

​We say that a positive integer $n$ is perfect if it is equal to the sum of its all proper divisors, i. e. $\sigma(n)-n = n$.

Find a necessary and sufficient condition for an even number $n$ to be perfect. Select ONE answer.

A

$n$ is a Mersenne prime.

B

$n$ is ​prime.

C

$n=2^{p-1}(2^p-1)$ for some prime $p$.

D

$n=2^{p-1}(2^p-1)$ for some prime $p$ such that $2^p-1$ is prime.