Alex's Adventures in Numberland - Alex Bellos [110]
This is an extremely large number. By comparison, the universe contains only 1080 elementary particles. Eventually, 6 pops up too, at a distance so far away that its position can only be conveniently described as a power of a power of a power of power:
The other numbers will also eventually appear, although – it must be stressed – with no sense of urgency. ‘The land is dying, even the oceans are dying,’ said Sloane with poetic flourish, ‘but one can take refuge in the abstract beauty of sequences like Dion Gijswijt’s A090822.’
As well as paying serious attention to prime numbers, the Greeks were even more enthralled by what they called perfect numbers. Consider the number 6: the numbers that divide it – its factors – are 1, 2 and 3. If you add 1, 2 and 3, voilà, you get 6 again. A perfect number is any number, like 6, that is equal to the sum of its factors. (Strictly speaking, 6 is also a factor of 6, but in discussions of perfection it only makes sense to include the factors of a number less than the given number.) After six, the next perfect number is 28 because the numbers that divide it are 1, 2, 4, 7 and 14, the sum of which is 28. Not only the Greeks, but Jews and Christians too attached cosmological significance to such numerical perfection. The ninth-century Benedictine theologian Rabanus Maurus wrote, ‘Six is not perfect because God has created the world in 6 days; rather, God has perfected the world in 6 days because the number was perfect.’
The practice of adding the factors of a number leads to the most whimsical concepts in maths. Two numbers are amicable if the sum of the factors of the first number equals the second number, and if the sum of the factors of the second number equals the first. For example, the factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Added they equal 284. The factors of 284 are 1, 2, 4, 71 and 142. Together they make 220. Sweet! The Pythagoreans saw 220 and 284 as symbols of friendship. During the Middle Ages talismans with these numbers were made, to promote love. One Arab wrote that he tried to test the erotic effect of eatg something labelled with the number 284, while a partner was eating something labelled 220. It was only in 1636 that Pierre de Fermat discovered the second set of amicable numbers: 17,296 and 18,416. Because of the advent of computer processing, more than 11 million amicable pairs are now known. The largest pair has more than 24,000 digits each, which makes them tricky to write on a slice of baklava.
In 1918 the French mathematician Paul Poulet coined the term sociable for a new type of numerical friendship. The five numbers listed below are sociable because if you add up the factors of the first one, you get the second. If you add up the factors of the second, you get the third. If you add up the factors of the third, you get the fourth, the factors of the fourth give you the fifth, and the factors of the fifth get you back to where you started: they add up to the first:
12,496
14,288
15,472
14,536
14,264
Poulet discovered only two chains of sociable numbers – the five numbers above and a less exclusive gang of 28 numbers beginning with 14,316. The next set of sociable numbers was discovered by Henri Cohen, but not until 1969. He found nine sociable chains of just four numbers each, of which the chain with the lowest values is 1,264,460, 1,547,860, 1,727,636 and 1,305,184. Currently, 175 chains of sociable numbers are known, and almost all are chains of four numbers. None are chains of three (particularly poetic, since we all know that three’s a crowd, and a group of four is much more sociable). The longest chain remains Poulet’s 28, which is curious, as 28 is also a perfect number.
It was the Greeks who worked out an unexpected link between perfect numbers and prime numbers, which led to many further numerical adventures. Consider the sequence of doubles starting at 1:
(A79) 1, 2, 4, 8, 16…
In The Elements, Euclid showed that whenever the sum of