# Tagged: Euler

## Problem 496

Prove that if $2^n-1$ is a Mersenne prime number, then
$N=2^{n-1}(2^n-1)$ is a perfect number.

On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.

## Beautiful Formulas for pi=3.14…

The number $\pi$ is defined a s the ratio of a circle’s circumference $C$ to its diameter $d$:
$\pi=\frac{C}{d}.$

$\pi$ in decimal starts with 3.14… and never end.

I will show you several beautiful formulas for $\pi$.