How Does Perfect Numbers and Mersenne Primes - Numberphile Work?

Perfect numbers and Mersenne primes are special kinds of numbers that follow some neat rules, like a secret club for certain numbers.

Imagine you have a group of friends who always share cookies equally. If someone has 6 cookies, they can split them into groups of 2 or 3, both of which divide evenly into 6. This makes 6 a perfect number because its proper divisors (1, 2, and 3) add up to the number itself.

Now, Mersenne primes are like super special prime numbers, numbers that only have two factors: 1 and themselves. These primes look like $2^n - 1$, where n is also a prime. For example, when n = 3, you get $2^3 - 1 = 7$, which is a Mersenne prime.

There's a cool link between perfect numbers and Mersenne primes! If you take a Mersenne prime and use this formula:

\frac{(2^n - 1) \times (2^{n-1})}{2}

you get an even perfect number. So, if $2^3 - 1 = 7$ is a Mersenne prime, then multiplying it by 2^{2} = 4, you get $7 × 4 = 28$. And 28 is a perfect number!

It’s like having a special recipe that makes new friends in the perfect number club, every time you find a Mersenne prime, you might also find a perfect number!