Perfect Numbers

The first question that comes to mind, when we talk about perfect numbers is, “What is a perfect number?” So, let’s answer that first.

Any number that can be written as the sum of all of its proper divisors (other than itself) is called a Perfect Number.

In about 300 BC Euclid showed that if 2p−1 is prime then (2p−1)2p−1 is a perfect number. Philo of Alexandria in his first-century book “On the creation” mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect.

The smallest Perfect Number is 6. Now, we know that the number 6 is divisible by 1, 2 & 3.

Now, 1+2+3 = 6

Next up, we have the number 28. Let’s check for the divisors of 28 and we have 1,2,4,7 & 14.

Now, 1+2+4+7+14 = 28

There’s a bit of a gap before we find the next perfect number. It is 496. Let’s check for the divisors of 496. They are, 1,2,4,8,16,31,62,124 & 248.

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

As we can observe, there’s only one perfect number between 1 & 10, only one perfect number between 10 & 100, only one perfect number between 100 & 1000 and there’s only one perfect number between 1000 & 10000. And that is 8128 (have fun with the calculations).

As you might have observed, all these perfect numbers are even. Interestingly, we don’t know if odd perfect numbers exist. All the perfect numbers we have found are even. Even with all our supercomputers, we have only been able to find 49 perfect numbers so far. We don’t yet know if there are infinitely many perfect numbers either.

Apparently, the largest known perfect number has one million seven hundred and ninety one thousand digits, “only”.

All the known Perfect Numbers are of the form:

, where
is a prime number, known as a Mersenne Prime.

Now, let us try to prove that any number of this form will always be a perfect number.

Let us take an example first: 496. It’s proper factors include: 1,2,4,8,16,31,62,124 & 248.

Let us also consider 496 among the factors for this proof, thus the sum we will get should be double the value of 496.

Thus, we have, Sum = 1+2+4+8+16+31+62+124+248+496

= 1+2+4+8+16+31*1+31*2+31*4+31*8+31*16

= 1(1+2+4+8+16)+31(1+2+4+8+16)

=(1+31)(1+2+4+8+16)

= 32*31

= 992

=2*496

Now, let us prove it for the generalised case.

The sum of the factors will be:

Which can re-written as:

Thus, we can take this sum common from both the terms

Thus, we will be left with:

Now, let’s try to figure out the value of this sum.
Let us assume that:

So, the sum of the factors becomes:

Now, since we had included the number itself in the factors, to find the sum of proper factors only, we divide the sum found by 2.

Thus we get, Sum of proper factors=

Hence, we have proven that any number of the above form is a perfect number.