Opinion | Finding amicable pairs: Sparse, but friendly

Amicable pairs of numbers are bound by a rule—which is that the sum of each number’s respective factors should be the other number in the pair

Yet again, it’s hard, in a time of political dustups and uncertainty, to focus on things mathematical for this column. In a time, that is, when such dustups dominate the news and our thoughts.

I refer, as you no doubt have guessed, to this week’s news about what was once India’s northernmost state, Jammu and Kashmir. What’s a mathematics dilettante to write about when everyone around is up to their eyes in political arguments, with opinions so entrenched that they will scorn and abuse anyone who disagrees? When such opinions carve out chasms between even otherwise good friends?

That last is a deliberate mention. Especially because this week began with Friendship Day, 4 August, I’m going to experiment with a possibly subtle message here.

In ancient Greece, the mathematician Pythagoras and his followers knew of and discussed pairs of numbers that were bound by a particular rule. This rule, and the pairs they defined, so fascinated them that they even attributed to these numbers mystical and spiritual powers. Today, it might seem a little strange, unless you’re a numerologist or an astrologer, to think of mere numbers in this way. But perhaps that gives you an idea of just how intrigued the Pythagoreans were with these numbers.

And what’s the rule I am referring to? Try this exercise. Choose two numbers at random. For each of them, list its factors, excluding the number itself (these are called its “aliquot factors" or “proper divisors"). Then add these factors. For example, if you took the pair 48 and 75, you’d first add up the aliquot factors of 48 — 1, 2, 3, 4, 6, 8, 12, 16 and 24 — to get 75. Then do the same with 75 — factors 1, 3, 5, 15 and 25 — for a total of 49. But not 48.

They’re very close, but 48 and 75 don’t follow the rule about these pairs—which is that the sum of each number’s respective factors should be the other number in the pair. 48’s factors give us 75, but 75’s factors give us 49. Not good enough.

But there are pairs of numbers that do follow this rule. In the spirit of the day that began this week, think of them as friends—for each, in a sense, can be broken down and then put together differently to produce the other. You know, friends do things for each other, that sort of thing. And in fact, this is such an appealing idea that mathematicians actually use it for these pairs. If they don’t quite refer to them as “friends", they call such pairs “amicable numbers".

And if 48 and 75 don’t make up such a pair, there are, indeed, plenty of others that do. The smallest amicable numbers are 220 and 284. Check:

Factors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Add them up to get 284.

Factors of 284: 1, 2, 4, 71, 142. Add them up to get 220.

As an example of such pairs, 220 and 284 were actually known to the Pythagoreans in Greece. Arab mathematicians of the 9th and 10th centuries AD didn’t just know about these pairs, one even found a formula that could generate some of them. Thabit Ibn Qurra’s (836-901AD) formula is relatively simple, but I won’t press it on you here, except to say that it involves prime numbers and powers of 2. Run a few calculations following the formula and voilà! An amicable pair pops out.

But not too many. Ibn Qurra’s method produces the smallest amicable pair, 220 and 284. It also produces the amicables 17,296 and 18,416, and one still larger pair. But that’s it. Ibn Qurra’s formula is good for only those three pairs. In the 18th century, Leonhard Euler studied these numbers and came up with his own formula, a generalization of Ibn Qurra’s—yet it produced only two more pairs. Yet, it’s a measure of the formidable, remarkable mathematician Euler that, formula or not, he actually discovered 58 more pairs of amicable numbers.

Now, before Euler, other renowned mathematicians like Pierre Fermat and René Descartes had also explored the delights of amicable numbers. So it is nothing short of astonishing that all of them, and all the great Arabs and Greeks before them, overlooked the second smallest pair, 1,184 and 1,210. It’s still more astonishing that that pair was first found by an Italian teenager called Nicolò Paganini (no, not the virtuoso composer), and only in 1866. Perhaps the most astonishing of all is that this Paganini is otherwise forgotten: as far as I can tell, the 1,184/1,210 pair is the only mark of any kind that he left on history.

Given that 220/284 is the smallest amicable pair, and the next smallest is 1,184/1,210, you might guess that amicable numbers are not exactly plentiful, and you’d be right. Here, increasing in size, are the first five pairs: 220 and 284, 1,184 and 1,210, 2,620 and 2,924, 5,020 and 5,564, 6,232 and 6,368. In fact, that’s the lot of them under 10,000: just five pairs. There are 13 pairs under 100,000 and 42 under a million.

They are sparse, but they seem to keep showing up without end. (“Seem", because we don’t yet know if there’s an infinity of them.) Euler knew of just 60 in the 18th century. But, as you can imagine, the age of the computer has allowed us to find many, many more. To give you an idea of that explosion: by 1946, we knew of 390 amicable pairs. But by 2007 that number had risen to nearly 12 million. The computers didn’t stop there either: today there are about 12 billion amicable pairs we know of. There’s even an ongoing worldwide search for them that you can join here: bit.ly/2KnUdTa. Don’t expect any significant prizes for finding new pairs, though, apart from recognition and gratitude from mathematicians around the world. (Though that may be significant enough for some.)

Yet, even though we know of so many amicable pairs and expect to keep finding more without limit, another mathematician actually showed just how sparse they really are. This was the great Paul Erdös (bit.ly/33kKg0A). In a 1955 paper he wrote as a 50th birthday present to the Hungarian mathematician László Kalmár, Erdös proved that the density of amicable pairs is actually zero. That is, as you get to truly large numbers, the ratio of amicable pairs to all the numbers you’ve traversed so shrinks as to be effectively non-existent. Or, to put it another way: however tiny a fraction you pick, we can always find a number so that the density of amicable pairs below that number is less than your fraction. Not a bad 50th birthday present, if you ask me.

Amicables are a special case of sociable numbers, or groups of number friends. Say you have three numbers, and the respective lists of their aliquot factors. If the sum of the first list is the second number, and the sum of the second is the third number, and the sum of the third is the first number—well, what you’d have is a sociable trio. You’d also have something nobody has yet found, so get ready for a deal of mathematical recognition and gratitude. That’s right: we know of some 12 billion amicable pairs, and we know of over 5,000 sociable quadruples—but we know not a single sociable trio.

And what about larger groups of sociable numbers? The numbers fall off dramatically. There’s one known five-member group, five six-member groups, four eight-member ones, one nine-member group and … wait for it … one 28-member group, whose smallest member is 14,316. That’s it.

There’s plenty more to savour about all these incredibly friendly numbers, which I’ll leave you to discover. But I wouldn’t blame you for wondering, by now, what it’s all about. What can we do with these numbers, what do they teach us?

The real answer is that they are yet another example of that phenomenon that excites any self-respecting mathematician: patterns in numbers. And in digging into these particular patterns, we learn ever more about the properties of numbers, the relationships between them, the power of certain mathematical functions, on and on. That pursuit of knowledge about numbers is valuable by itself, but it can also pay unexpected dividends in other aspects of our lives. (What makes your credit card transactions secure? How can we detect fraudulent accounting data? Thank number theorists.)

There’s no end, really, to how deep and wide in mathematics you can journey. For now, I’ll leave you to ruminate on the very idea of numbers that are friendly, that reach out to help each other. In a time of divides and disagreement, perhaps we can all use a dose of thoughts on friendship: numbers and otherwise, subtle or not.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun

(Editor's note: The factors of 48 add up to 76, not 75.)

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