Take a look at the sums of the first few powers of two.

1 + 2 = 3

1 + 2 + 4 = 7

1 + 2 + 4 + 8 = 15

1 + 2 + 4 + 8 +16 = 31

1 + 2 + 4 + 8 + 16 + 32 = 63

1 + 2 + 4 + 8 + 16 + 32 + 64 = 127

Interestingly, a good number of these sums are prime numbers. And this is no fluke. Farther down the line we get several more prime numbers. In fact, these primes are so prevalent we have a name for them: Mersenne Primes.

Aside from being the sum of powers of two, these prime numbers are more commonly expressed as one less than a power of two. This should be unsurprising, as a the sum of the first n powers of two is 2^(n+1) - 1, hence is essentially the exact same thing.

Taking a closer look at these primes, we can see that they appear only for prime quantities of m in 2^m - 1. Note that the converse does not apply; prime values of m (sadly) does not always give a Mersenne Prime.

Perfect Numbers

Perfect numbers are numbers that can be expressed as the sum of their proper divisors (not including the numbers itself). An example would be 6, the smallest perfect number. 
6 = 1 + 2 + 3

In relation to Mersenne primes, Euclid showed that for every such prime, there is corresponding perfect number. More formally:

For every Mersenne Prime 2^n - 1,

there exists an even perfect number

The Euclid-Euler Theorem

The Euclid-Euler theorem is expresses the correlation between Mersenne Primes and Perfect Numbers both ways. For this article, we will only be showing that there exists a Perfect Number for every Mersenne Prime.

We can take a look at a specific example, and make generalizations from there. Take Mersenne Prime 31, and it’s corresponding Perfect Number 496. The proper divisors of the latter are

496 => 248, 124, 62, 31, 16, 8, 4, 2, 1

From this, we can see that it includes the Mersenne prime (as per the relationship) and all powers of two less than said prime.

248, 124, 62, 31, 16, 8, 4, 2, 1

Furthermore, the divisors to the left of the Mersenne prime are just obtained by multiplying it by the other powers of two. This is of course excluding those that would give the Perfect Number itself (31*16=496) and duplicates (31*1=31).

248 = 31*8, 124 = 31*4, 62 = 31*2, 31, 16, 8, 4, 2, 1

We can therefore say that a Perfect Number would have three ‘categories’ of factors.

1.) The powers of two less than said Mersenne Prime

2.) The products of 1.) and the Mersenne Prime itself (which would include the actual Mersenne Prime)

Generalization

For any Perfect number, which recall can be expressed as

We can simply take the sum of these two categories:

1.) The powers of two less than said Mersenne Prime

1+2 + … + 2^(n-1) = 2^n - 1(sum of a geometric sequence)

2.) The products of 1.) and the Mersenne Prime itself

(2^n - 1)(2^n - 1)

Notice here that a Mersenne Prime is actually equal to the sum of the powers of two less than it.

Now all we have to do is get the sum of all three quantities, and show that it indeed equals twice the Perfect Number we started with. We can easily do this by factoring out 2^n - 1, which leave us with:

And dividing this by two indeed gives the initial Perfect Number:

As you can see, we arrived at our initial expression. The expression also equals the twice the sum of its factors, which makes it a Perfect Number. Ergo, we have proven the relation that for every Mersenne Prime, there exists a perfect number, which is the Mersenne Prime multiplied by a power of two.

The Powers of Two

Due to their simplicity, the powers of two find themselves in all sorts of places in number theory. By extent, they also connect these different fields. The ease by which they can be manipulated, and their elegant properties allow us to associate two seemingly unrelated sets of numbers. In the same way, our hunger for truth and justice connect us, even in these dire times. Mathematics connects. And it is this connection we share that gives hope to strive for a better tomorrow.