The Number Mysteries: A Mathematical Odyssey through Everyday Life. Marcus Sautoy du
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Название: The Number Mysteries: A Mathematical Odyssey through Everyday Life
Автор: Marcus Sautoy du
Издательство: HarperCollins
Жанр: Математика
isbn: 9780007362561
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How long would it take to write a list of all the primes?
Anyone who tried to write down a list of all the primes would be writing for ever, because there are infinitely many of these numbers. What makes us so confident that we’d never come to the last prime, that there will always be another one waiting out there for us to add to the list? It is one of the greatest achievements of the human brain that with just a finite sequence of logical steps we can capture infinity.
The first person to prove that the primes go on for ever was a Greek mathematician living in Alexandria, called Euclid. He was a student of Plato’s, and he also lived during the third century BC, though it appears he was about 50 years older than the librarian Eratosthenes.
To prove that there must be infinitely many primes, Euclid started by asking whether, on the contrary, it was possible that there were, in fact, a finite number of primes. This finite list of primes would have to have the property that every other number can be produced by multiplying together primes from this finite list. For example, suppose that you thought that the list of all the primes consisted of just the three numbers 2, 3 and 5. Could every number be built up by multiplying together different combinations of 2s, 3s and 5s? Euclid concocted a way to build a number that could never be captured by these three prime numbers. He began by multiplying together his list of primes to make 30. Then—and this was his act of genius—he added 1 to this number to make 31. None of the primes on his list, 2, 3 or 5, would divide into it exactly. You always got remainder 1.
Euclid knew that all numbers are built by multiplying together primes, so what about 31? Since it can’t be divided by 2, 3 or 5, there had to be some other primes, not on his list, that created 31. In fact, 31 is a prime itself, so Euclid had created a ‘new’ prime. You might say that we could just add this new prime number to the list. But Euclid can then play the same trick again. However big the table of primes, Euclid can just multiply the list of primes together and add 1. Each time he can create a number which leaves remainder 1 on division by any of the primes on the list, so this new number has to be divisible by primes not on the list. In this way Euclid proved that no finite list could ever contain all the primes. Therefore there must be an infinite number of primes.
Although Euclid could prove that the primes go on for ever, there was one problem with his proof—it didn’t tell you where the primes are. You might think that his method produces a way of generating new primes. After all, when we multiplied 2, 3 and 5 together and added 1, we got 31, a new prime. But it doesn’t always work. For example, consider the list of primes 2, 3, 5, 7, 11 and 13. Multiply them all together: 30,030. Now add 1 to this number: 30,031. This number is not divisible by any of the primes from 2 to 13, because you always get remainder 1. However, it isn’t a prime number since it is divisible by the two primes 59 and 509, and they weren’t on our list. In fact, mathematicians still don’t know whether the process of multiplying a finite list of primes together and adding 1 infinitely often gives you a new prime number.
There’s a video available of my football team in their prime number kit explaining why there are infinitely many primes. Visit http://bit.ly/Primenumbersfootball or use your smartphone to scan this code.
Why are my daughters’ middle names 41 and 43?
If we can’t write down the primes in one big table, then perhaps we can try to find some pattern to help us to generate the primes. Is there some clever way to look at the primes you’ve got so far, and know where the next one will be?
Here are the primes we discovered by using the Sieve of Eratosthenes on the numbers from 1 to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
59, 61, 67, 71, 73, 79, 83, 89, 97
The problem with the primes is that it can be really difficult to work out where the next one will be, because there don’t seem to be any patterns in the sequence that will help us to help locate them. In fact, they look more like a set of lottery ticket numbers than the building blocks of mathematics. Like waiting for a bus, you can have a huge gap with no primes and then suddenly several come along in quick succession. This behaviour is very characteristic of random processes, as we shall see in Chapter 3.
Apart from 2 and 3, the closest that two prime numbers can be is two apart, like 17 and 19 or 41 and 43, since the number between each pair is always even and therefore not prime. These pairs of very close primes are called twin primes. With my obsession for primes, my twin daughters almost ended up with the names 41 and 43. After all, if Chris Martin and Gwyneth Paltrow can call their baby Apple, and Frank Zappa can call his daughters Moon Unit and Diva Thin Muffin Pigeen, why can’t my twins be 41 and 43? My wife was not so keen, so these have had to remain my ‘secret’ middle names for the girls.
Although primes get rarer and rarer as you move out into the universe of numbers, it’s extraordinary how often another pair of twin primes pops up. For example, after the prime 1,129 you don’t find any primes in the next 21 numbers, then suddenly up pop the twin primes 1,151 and 1,153. And when you pass the prime 102,701 you have to plough through 59 non-primes, and then the pair of primes 102,761 and 102,763 suddenly appear. The largest twin primes discovered by the beginning of 2009 have 58,711 digits. Given that it only takes a number with 80 digits to describe the number of atoms in the observable universe, these numbers are ridiculously large.
But are there more beyond these two twins? Thanks to Euclid’s proof, we know that we’re going to find infinitely many more primes, but are we going to keep on coming across twin primes? As yet, nobody has come up with a clever proof like Euclid’s to show why there are infinitely many of these twin primes.
At one stage it seemed that twins might have been the key to unlocking the secret of prime numbers. In The Man Who Mistook His Wife for a Hat, Oliver Sacks describes the case of two real-life autistic savant twins who used the primes as a secret language. The twin brothers would sit in Sacks’s clinic, swapping large numbers between themselves. At first Sacks was mystified by their dialogue, but one night he cracked the secret to their code. Swotting up on some prime numbers of his own, he decided to test his theory. The next day he joined the twins as they sat exchanging six-digit numbers. After a while Sacks took advantage of a pause in the prime number patter to announce a seven-digit prime, taking the twins by surprise. They sat thinking for a while, since this was stretching the limit of the primes they had been exchanging to date, then they smiled simultaneously, as if recognizing a friend.
During their time with Sacks, they managed to reach primes with nine digits. Of course, no one would find it remarkable if they were simply exchanging odd numbers or perhaps even square numbers, but the striking thing about what they were doing is that the primes are so randomly scattered. One explanation for how they managed it relates to another ability the twins had. Often they would appear on television, and impress audiences by identifying that, for example, 23 October 1901 was a Wednesday. Working out the day of the week from a given date is done by something called modular or clock arithmetic. Maybe the twins discovered that clock arithmetic is also the key to a method that identifies whether a number is prime.
If you take a number, say 17, and calculate 217, then if the remainder when you divide this number by 17 is 2, that is good evidence that the number 17 is prime. This test for primality is often wrongly attributed to the Chinese, and it was the seventeenth-century French mathematician Pierre de Fermat who proved that if the remainder isn’t 2, then that certainly implies that 17 is not prime. In general, if you want to check that p is not a prime, then calculate 2p СКАЧАТЬ