A friend is another self, like 220 and 284.
— Attributed to Pythagoras (c575 - c495 BC).

Mathematicians aren't renowned for being the most gregarious of people. To get to grips mentally with all that precise logic they often require to be left well alone. Andrew Wiles became a virtual recluse for seven years as he plotted his successful final onslaught on Fermat's Last Theorem, only emerging from his attic occasionally for air and to give the odd lecture. In 1940, the similarly-named French mathematician André Weil developed his most important contributions to algebraic geometry while in prison in Rouen for avoiding enlistment into the army. Another French mathematician, André Bloch, did all his best work during his 31 years as an inmate at a mental institution¹.

They may not have much in the way of social lives, but, to compensate, mathematicians have created a whole world of imaginary friends in the numbers they study. We'll describe these, but first, we have to do a little maths...

The Joy of Division

Mathematicians study the theory of numbers within a branch of the subject known, not surprisingly, as number theory. One interesting property of integers (whole numbers) is whether they can be divided exactly. As you probably learned at school, some numbers divide into lots of different factors, whereas those we know as prime numbers can be divided only by themselves or 1.

Now, each integer has a unique set of numbers into which it divides, and mathematicians have for a long time been interested in these. If we take, say, the number 42, its divisors are 1, 2, 3, 6, 7, 14, 21 and 42. Those of the number 12 are 1, 2, 3, 4, 6 and 12.

So, what can we do with these divisors? Well, we can count them, of course, but we find some of the most interesting mathematical results if we add them up. This sum of divisors is commonly known as the divisor function, and we give it the Greek letter sigma (σ), so:

σ(42) = 1 + 2 + 3 + 6 + 7 + 14 + 21 + 42 = 96, and
σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28

The last number in this divisor list is the number itself, but sometimes we don't count this. If not, then we call these proper divisors and when we add them up, we call this the restricted divisor function – some call it the aliquot sum² – and we give it the letter s. So:

s(42) = 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54, and
s(12) = 1 + 2 + 3 + 4 + 6 = 16

So, what can we do with these new numbers we have found?

Making Friends

Some integers are related to their divisor sum in similar ways. Those which share a common ratio between the number and divisor function are known as friendly numbers. The lowest number to have a friend is 6, who is a friend of 28. 6 has the divisors 1, 2, 3 and 6, which add up to 12. So its divisor sum is exactly twice as large as the number itself. 28 has the factors 1, 2, 4, 7, 14 and 28, which add up to 56, which again is twice 28.

6 and 28 are known as a friendly pair. You can look for more friends of 6 and 28, but you won't find another until you get to 496, and in fact there's a fourth at 8,128. These particular friends are in fact a special set of numbers known as perfect numbers.

Another well-known friendly pair is 30 and 140 (which have three further friends in 2,480, 6,200 and 40,640). 80 and 200 are also the best of mates.

Sometimes, as with Stan Laurel, a number can have a much larger friend. 24's closest pal is 91,963,648. Both numbers have a divisor sum which is 2½ times their value.

But alas, there are many which remain friendless. Of the numbers from 1 to 9, only the number 6 has any friends. The others are known as solitary numbers. Mathematicians have managed to prove that these numbers have no friends, but they can't yet prove this for all the currently friendless numbers. 10 is the lowest number for which they're not sure. They think it's solitary, but if you can prove it, you will achieve fame and fortune (well, just fame, actually – and only in mathematical circles). You will equally earn respect if you disprove the mathematicians: just do a bit of matchmaking and find 10 a friend.

Amicable Numbers

Numbers don't need to be fully-fledged friends, they can also get on with each other amicably. Each of an amicable pair of numbers has a reduced divisor sum which is equal to the other number in the pair.

The smallest amicable pair is 220 and 284. It works as follows:

s(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284;
s(284) = 1 + 2 + 4 + 71 + 142 = 220.

Other amicable pairs include 1,184 & 1,210, and 2,620 & 2,924.

Amicable numbers have been known about since ancient times. Greek lovers would present each other with talismans engraved with each of an amicable pair of numbers, representing the bond between them. They would often try to derive these numbers from their own personal statistics: their birth dates, horoscopes, or even their height.

Since then, some of the most famous mathematicians, including Fermat, Descartes and Euler, have spent time finding new amicable pairs. It's still going on, too; in 2005, Paul Jobling discovered an amicable pair of numbers each of which was 24,073 digits long, while he was being held in a queue to speak to a call-centre³.

Sociable Numbers

Amicable numbers only occur in pairs, but it's possible to have a larger group of numbers where the restricted divisor sum of the first leads to the next, and so on, before it returns to the first in the list. Groups like this are known as sociable numbers, but they're all quite large. One sociable group of four is {1,264,460, 1,547,860, 1,727,636 and 1,305,184}. There are smaller numbers in this group of five: {12,496, 14,288, 15,472, 14,536 and 14,264}. Mathematicians haven't yet found a social threesome.

Practical Applications

Mathematics is one of the most fundamental of disciplines. If we ever leave this planet and colonise new worlds, maybe in distant galaxies, we may find that all our scientific knowledge – our chemistry, biology and physics – operates in a completely different way in another part of the universe. Mathematics, on the other hand, is universal. We can be sure that our arithmetic, geometry and even our number theory will still be relevant wherever it is used.

One thing we might well hope for in another world, though, is a practical application for friendly numbers, because we sure as hell can't find one in ours.