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Just For Fun

Perfect numbers

“If you look for perfection, you’ll never be content”

There are very few things that are perfect in the world. But in maths, there are perfect numbers.

What is a perfect number?

To describe what a perfect number is, we first need to know what a factor is.

The factors of a number are numbers that you can divide the number by and get a whole number result.

For example, 2 is a factor of 10 because 10 ÷ 2 = 5 (which is a whole number).

But 4 is not a factor of 10 because 10 ÷ 4 = 2.5 (which is not a whole number).

For more about factors, read my previous blog or try our worksheet.

A perfect number is a number which is equal to the sum of its factors, excluding the number itself. This is also known as a number’s proper divisors. The sum of a number’s proper divisors is also known as its Aliquot sum.

For example, take the number 12.

Here are the proper divisors of 12:

1, 2, 3, 4, 6

Add these together: 1 + 2 + 3 + 4 + 6 = 16. Close, but not quite perfect.

But, take a look at the proper divisors of 28:

1, 2, 4, 7, 14

1 + 2 + 4 + 7 + 14 = 28. Perfection! So 28 is a perfect number.

How many perfect numbers are there?

The truth is, we don’t know… What we do know is that they are very rare, and there are very few perfect numbers that are feasibly writable on a page. What I mean by this, is that there are only 12 perfect numbers that have fewer than 300 digits!

To date, only 51 perfect numbers have been discovered, but we do not know if this list is exhaustive or not. It has not been proven whether there are infinitely many perfect numbers – only once this is proven will we know the answer for sure.

The largest perfect number discovered (by supercomputers) has almost 50 million digits. To put the enormity of this number in to context, if you wanted to write this number on paper, assuming that you could write 2 digits on the page every second (which may be optimistic) and never took any breaks, it would take you over 9 months to write the number down.

Since those numbers are unfathomably large, let’s focus on some smaller perfect numbers. Here are the smallest known perfect numbers:

6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8,128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064 

The next smallest perfect number is 33,550,336. The sequence of perfect numbers is sequence A000396 on the OEIS.

You may spot a few things that these numbers have in common. Notably they are all even numbers. No odd perfect numbers have yet been discovered, and it is not known whether there are any (although it would be fair to assume that they are very unlikely given the huge numbers that have been tested already). Even more specifically, all known perfect numbers end in either 6 or 8.

Another very interesting trait of the known perfect numbers is that they are all triangular numbers.

Abundant and deficient numbers

Piggybacking on the concept of a perfect number are abundant and deficient numbers. You may be able to work out what these are based on their names. If a perfect number is a number where the sum of the number’s proper divisors is equal to the number itself, an abundant number is a number where the sum of its proper divisors is larger than the number and a deficient number is a number where the sum of its proper divisors is smaller than the number.

For example, take the numbers 18 and 22.

Here are the proper divisors of 18:

1, 2, 3, 6, 9

1 + 2 + 3 + 6 + 9 = 21 > 18 so 18 is an abundant number

Here are the proper divisors of 22:

1, 2, 11

1 + 2 + 11 = 14 < 22 so 22 is a deficient number

This means that every positive integer is either perfect, abundant or deficient. There are infinitely many abundant and deficient numbers. And, as you would expect, there are many more deficient numbers than abundant numbers and perfect numbers.

For example, looking at all positive integers from 1 to 100, 76 of them are deficient, 22 are abundant and only 2 are perfect.

As an exercise, can you find all 22 abundant numbers between 1 and 100? I’ve already given away that 18 is one of them. Email your answers to sam@metatutor.co.uk and I’ll tell you if you’re right!

I hope you found this blog interesting. If you are in Bristol and have a son or daughter who need help with any aspect of mathematics, book in a free taster session.

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