Molly Roberts

Do numbers make friends? Well, everyone who watched the Numberjacks on TV as a child knows the answer is most certainly YES. A gang of numbers hang out inside a sofa, solve maths problems, and sort out trouble caused by the likes of the Puzzling Puzzler, or the terrifying Spooky Spoon. Whilst sentient crime-fighting numbers living in a London couch might be a bit of a stretch of the imagination, even the most seasoned of mathematicians would have to agree that numbers do indeed have the capacity to get on with one another. In part 2 of the numbers with cool names series, we’ll be delving into what it means for a number to be amicable, friendly, or even sociable, with other numbers. Who knew you could rigorously prove that the number 3 is an introvert?

Amicable Numbers

Remember perfect numbers from part 1? Well, amicable numbers are pairs of numbers where the proper divisors of the first sums to give you the second, and vice versa. This is a similar concept to perfect numbers – just like before, you calculate proper divisors, and then sum them up.

I first read about amicable numbers in the book ‘Fermat’s Last Theorem’ by Simon Singh when I was twelve years old, and I remember thinking they were cool. So cool, in fact, that I seriously considered making a set of extremely nerdy friendship bracelets with one amicable number on one and its pair on the other. How embarrassing, you might think. Or, perhaps you are a nerd like me and find the idea of being interlinked with another person via the use of irrevocably connected numbers is kind of attractive. Either way, you started reading this article so you must have at least a small amount of interest in cool numbers too!

The first example of a pair of amicable numbers is 220 and 284. 220 has proper divisors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. 284 has proper divisors 1, 2, 4, 71, and 142.

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 = 284

1 + 2 + 4 + 71 + 142 = 220

Hence 220 and 284 are amicable.

Another, considerably more impressive example, was discovered by the Iranian mathematician Muhammad Baqir Yazdi: 9,363,584 and 9,437,056. Wow! Can you imagine working that out by hand? I certainly can’t…

As it turns out, there are literally millions of pairs of amicable numbers. There is even a website which tells you how many pairs there are with a certain number of digits, which you can find here. As of September 2021, there have been 1,227,184,297 pairs found, both by hand and by computer. 

A general formula for finding some of these pairs has been credited to Iraqi mathematician Thābit ibn Qurra. More detail on the history of amicable numbers can be found in a 2003 article by Patrick J. Costello called ‘New Amicable Pairs of Type (2, 2) and Type (3, 2)’, a preview of which can be seen here.

If you’re still with me at this point, I’d suggest having a look at some of the other known pairs of amicable numbers. Which pair would you choose if you were to create a couple of maths friendship bracelets? And perhaps most importantly which friend would you give the second bracelt to?

Sociable Numbers

Sociable numbers are to the integers what friend groups are to human beings. They are an extension of the concepts of perfect and amicable numbers to cycles of numbers. The easiest way to explain this is with an example.

Take the numbers 12496, 14288, 15472, 14536, and 14264. If you work out the divisors of each number you will see that the divisors of 12496 sum to 14288, the divisors of 14288 sum to 15472, the divisors of 15472 sum to 14536, the divisors of 14536 sum to 14264, and finally the divisors of 14264 sum to 12496, completing the cycle. In fact, since there are five numbers in this cycle, it is known as a set of sociable numbers of order 5.

The cycle above was discovered in 1918 by the Belgian mathematician Paul Poulet. The second cycle he discovered had order 28, which is entirely too long to write down, and rather mind boggling to think about.

Now, you might recognise straight away that both perfect numbers and amicable numbers are types of sociable numbers: perfect numbers are defined as a set of sociable numbers of order 1, and amicable numbers are defined as a set of sociable numbers of order 2. There are no known sets of sociable numbers of order 3.

If you’d like to investigate these fascinating numbers further, I suggest looking up some other sets of sociable numbers of different orders, or even trying to find the numbers involved in Poulet’s order 28 cycle. Maybe you can try to find a cycle that has the same size as your friendship group and then you can extend the friendship bracelet idea to everyone!

Friendly Numbers

Not to be confused with amicable numbers. Which I admit is very confusing, especially as these two terms often seem to be used interchangeably outside of maths. However, we are clever people and are not going to be confused. Hopefully.

Friendly numbers are described as two or more numbers which share the same abundancy.

What is abundancy, you may ask? Abundancy is the value obtained by summing a number n, and all its proper divisors, together, and then dividing this total by n. This concept is once again easier to explain by looking at some examples.

First, let’s look at 6. This has proper divisors 1, 2, and 3. Hence the abundancy is worked out by calculating the value of (1 + 2 + 3 + 6) / 6, which is equal to 2. Therefore, 6 has abundancy 2.

Let’s try another number, for example, 12. This has proper divisors 1, 2, 3, 4, and 6, and so has abundancy (1 + 2 + 3 + 4 + 6 + 12) / 12 = 7/3.

Now let’s look at the number 6 more closely. As you might remember from part 1, 6 is a perfect number. In fact, all perfect numbers have abundancy 2, since their proper divisors sum to their exact value. This means that all perfect numbers are friendly with each other, because they all have the same abundancy. In fact, only the perfect numbers have abundancy 2. Sound familiar? All the perfect kids at school hanging out solely with each other?

Well, never fear, because there are plenty of other friendship groups to be part of. For example, 30 and 140 are friendly, and so are 80 and 200. They might not be perfect, but that doesn’t mean they can’t be the best friends you’ve ever had.

Don’t want to talk to anyone? Maths has got you covered too. There are plenty of numbers who aren’t friendly with any others, i.e. their abundancy is unique to them. In fact, there are infinitely many of these numbers, which we call solitary numbers. For some simple examples, it’s relatively straightforward to show that every prime is a solitary number. 

Now, have a look at the non-perfect friendly groups I mentioned above. Can you prove that 30 and 140 are friendly with each other? How about 80 and 200? I’ll leave you to investigate…

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Answers

30 has proper divisors 1, 2, 3, 5, 6, 10, and 15. Hence it has abundancy (1 + 2 + 3 + 5 + 6 + 10 + 15 + 30) / 30 = 125. 140 has proper divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70. Hence it has abundancy (1 + 2 + 4 + 5 + 7 + 10 + 14 + 20 + 28 + 35 + 70 + 140) / 140 = 125.

80 has proper divisors 1, 2, 4, 5, 8, 10, 16, 20, and 40. Hence it has abundancy (1 + 2 + 4 + 5 + 8 + 10 + 16 + 20 + 40 + 80) / 80 = 9340. 200 has proper divisors 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, and 100. Hence it has abundancy (1 + 2 + 4 + 5 + 8 + 10 + 20 + 25 + 40 + 50 + 100 + 200) / 200 = 9340.

If you enjoyed this article on numbers with cool names, be sure to check out the other two in the series:

Numbers With Cool Names Part 1 – Happy, Lucky, Perfect

Numbers With Cool Names Part 3 – Weird, Sexy, Untouchable