# Perfect Numbers

An ancient riddle

In mathematics, especially when it comes to *number theory*, there are many basic things that we don’t know.

Simple questions that we can’t answer.

This can be realized by considering a few of the famous unanswered questions that we mathematicians would like to know the answer to.

A few of them are

- Are there infinitely many primes p such that p+2 is also prime?
- Are there infinitely many Mersenne primes?
- Can every even number greater than 2 be written as a sum of two primes?
- Are there infinitely many amicable numbers?
- Is the Collatz conjecture true?

These questions seem deceptively simple, but these are actually some of the deepest mysteries of the field and we don’t have a clue about how to even approach them. We are in desperate need of both knowledge and machinery.

This list is only the beginning of our embarrassing lack of knowledge. In this article, we shall add some more mysteries to the list.

# What we know

The theory of perfect numbers is one of the oldest in mathematics. It all started in about 300 BC when Euclid proved an amazing property about them.

Before getting into that though, we need to define them so that we are all on the same page.

Recall that a divisor of a number n is a number d such that d divides n. A proper divisor of n is defined as a divisor of n that is not n i.e. it must be strictly less than n.

Let’s consider the number 28.

28 is a divisor of 28 and 7 is a proper divisor of 28.

The proper divisors of 28 are 1, 2, 4, 7, and 14.

A perfect number is a natural (whole and positive) number such that the sum of its proper divisors is equal to itself.

For example, 6 is a perfect number because its proper divisors are 1, 2, and 3, and the sum of these is 6.

6 is the smallest perfect number and 28 is the next one. You can try adding the proper divisors of 28 listed above - you should get 28.

The next ones are 496 and 8128 and they get pretty big pretty fast.

We can also define perfect numbers using the sum of (all) the divisors.

That sum should be twice the number if it’s perfect. The sum of divisors of a number n is denoted σ(n) thus by that notation, a number n is perfect if and only if σ(n) = 2n e.g. 1+2+3+6 = 12 = 2 ⋅ 6.

Recall that two numbers are relatively prime if the only number dividing both of them is 1.

The sum of divisor function σ is what’s called an arithmetic function and one can prove that it is multiplicative i.e. if n and m are relatively prime, then σ(nm) = σ(n)σ(m).

The truth of this fact can be realized by simple combinatorics. Consider the number k = nm where n and m are relatively prime (i.e. n and m doesn’t share any non-trivial divisors). A divisor of k then must be of the form dc where d divides n and c divides m. Moreover, all the combinations of such numbers dc give us all the divisors of k. The sum of divisors of k is therefore

In about 300 BC Euclid showed the following important discovery.

Let’s prove this.

We are going to use the elegant proof that Euler discovered where he applied the sum of divisor function.

Before I state the proof, recall from my article about Mersenne primes that a partial sum of a geometric series has a closed-form expression. In particular,

Suppose that 2^p-1 is a prime number. By the above discussion, we have the following:

showing that m must be a perfect number.

Note that we have used the multiplicative property of σ (it’s clear that the two numbers are relatively prime because a power of 2 only has even divisors and the other is odd leaving it with no even divisors) as well as the closed-form of the geometric sum with base 2.

A small comment here: note that m must be even because of the power of 2 lying around.

Euclid was the first person to prove this fact. More than 2000 years of frustrations followed before Euler proved something astonishing.

Euler proved that all even perfect numbers are of this form.

That is, if *m* is an even perfect number, then m = 2^(p - 1)(2^p-1) for some prime p, and 2^p - 1 is prime.

I will give you a proof here:

Primes of this form are called Mersenne primes so what Euler showed was that there is a one-to-one correspondence between even perfect numbers and Mersenne primes.

A natural question is of course if there are infinitely many even perfect numbers and the answer is: we don’t know! Of course, an equivalent question is if there are infinitely many Mersenne primes which is a deep mystery.

Another “perfect” riddle is: are there any odd perfect numbers? And once again, I am sorry to disappoint you but we don’t know that either.

We do know that if there exists an odd perfect number, then it has to satisfy a parade of hard constraints, so by a heuristical argument, the answer is: we don’t think so… But that is of course not good enough.

We are also pretty certain that there are infinitely many Mersenne primes and therefore also infinitely many (even) perfect numbers, but a proof of that seems far away. We are waiting for a new Euler to prove it.

# Why Study Perfect Numbers?

You might be wondering why we are interested in perfect numbers in the first place. Euclid was because they possess some kind of underlying dividing beauty, Euler was, probably, because it was a great challenge and he had an insight about using the sum of divisors function (or maybe he had only published 49 papers that year!). But why are we?

Well, the even perfect numbers are in some sense equivalent to Mersenne primes and Mersenne primes are used in online security, so we are interested in finding out more about them.

As mentioned above, we don’t know if we will ever run out of Mersenne primes.

At least we know *that *about our oil reserves but of course Mersenne primes do not pollute, so we are not leaving a mathematical ruin for the next generations if we use them all, so no need to have a bad conscience over that, if that should be the case!

But knowing more about perfect numbers would almost certainly require knowing more about the “sum of divisor” function which is closely related to the Riemann zeta function in which information about the distribution of the prime numbers is encoded.

And then of course: if a famous problem has survived for more than 2000 years, then it is because we need an extraordinary idea or theory and THAT is worth waiting for.

For now, though, I will sit back and wait — hoping that a new Euler will arrive in my lifetime — even though the odds are slim… at best.