The Number Mysteries
A Mathematical Odyssey through Everyday Life
By Marcus Du Sautoy
Palgrave Macmillan Copyright © 2011 Marcus du Sautoy
All rights reserved.
THE CURIOUS INCIDENT OF THE NEVERENDING PRIMES
1, 2, 3, 4, 5 ... it seems so simple: add 1, and you get the next number. Yet without numbers, we'd be lost. Arsenal vs. Manchester United—who won? We don't know. Each team scored lots of goals. Want to look something up in the index of this book? Well, the bit about winning the National Lottery is somewhere in the middle of the book. And the lottery itself? Hopeless without numbers. It's quite extraordinary how fundamental the language of numbers is to negotiating the world.
Even in the animal kingdom, numbers are fundamental. Packs of animals will base their decision to fight or flee on whether their group is outnumbered by a rival pack. Their survival instinct depends in part on a mathematical ability, yet behind the apparent simplicity of the list of numbers lies one of the biggest mysteries of mathematics.
2, 3, 5, 7, 11, 13 ... These are the primes, the indivisible numbers that are the building blocks of all other numbers—the hydrogen and oxygen of the world of mathematics. These protagonists at the heart of the story of numbers are like jewels studded through the infinite expanse of numbers.
Yet despite their importance, prime numbers represent one of the most tantalizing puzzles we have come across in our pursuit of knowledge. Knowing how to find the primes is a total mystery because there seems to be no magic formula that gets you from one to the next. They are like buried treasure—and no one has the treasure map.
In this chapter, we will explore what we do understand about these special numbers. In the course of this journey, we will discover how different cultures have tried to record and survey primes and how musicians have exploited their syncopated rhythm. We will find out why the primes have been used to try to communicate with extraterrestrials and how they have helped to keep things secret on the Internet. At the end of the chapter, I unveil a mathematical enigma about prime numbers that will earn you a million dollars if you can crack it. But before we tackle one of the biggest conundrums of mathematics, let us begin with one of the great numerical mysteries of our time.
WHY DID BECKHAM CHOOSE THE NUMBER 23 SHIRT?
When David Beckham joined the Real Madrid soccer team in 2003, there was a lot of speculation about why he'd chosen to play in the number 23 shirt. It was a strange choice, many thought, since he'd been playing in the number 7 shirt for England and Manchester United. The trouble was that on the Real Madrid team, the number 7 shirt was already being worn by Raúl, and the Spaniard wasn't about to move over for this glamour-boy from England.
Many different theories were put forth to account for Beckham's choice, and the most popular was the Michael Jordan theory. Real Madrid wanted to break into the American market and sell lots of replica shirts to the huge US population. But soccer is not a popular game in the States. Americans like baseball and basketball—games that end with scores like 100 to 98 and in which there's invariably a winner. What's the point of a game that goes on for 90 minutes and can end with no side scoring or winning?
With this theory in mind, Real Madrid did its research and found that the most popular basketball player in the world was definitely Michael Jordan, the Chicago Bulls' most prolific scorer. Jordan sported the number 23 shirt for his entire career. All Real Madrid had to do was put 23 on the back of a soccer shirt, cross their fingers, and hope that the Jordan connection would work its magic and that they would break into the American market.
Others suggested a more sinister theory. Julius Caesar was assassinated by being stabbed in the back 23 times. Was Beckham's choice for his back a bad omen? Still others thought that maybe the choice was connected with Beckham's love of Star Wars (Princess Leia was imprisoned in Detention Block AA23 in the first Star Wars movie). Or was Beckham a secret member of the Discordianists, a modern cult that reveres chaos and has a cabalistic obsession with the number 23?
But as soon as I saw Beckham's number, a more mathematical solution immediately came to mind. 23 is a prime number. A prime number is a number that is divisible only by itself and 1. 17 and 23 are prime because they can't be written as two smaller numbers multiplied together, whereas 15 isn't prime because 15 = 3 × 5. Prime numbers are the most important numbers in mathematics because all other whole numbers are built by multiplying primes together.
Take 105, for example. This number is clearly divisible by 5. So I can write 105 = 5 × 21. 5 is a prime number—an indivisible number—but 21 isn't: I can write it as 3 × 7. So 105 can also be written as 3 × 5 × 7. But this is as far as I can go: I'm down to the primes, the indivisible numbers from which the number 105 is built. I can do this with any number since every number is either prime and indivisible or not prime and divisible by smaller numbers multiplied together.
The primes are the building blocks of all numbers. Just as molecules are built from atoms, such as hydrogen and oxygen or sodium and chlorine, numbers are built from primes. In the world of mathematics, the numbers 2, 3, and 5 are like hydrogen, helium, and lithium. That's what makes them the most important numbers in mathematics. But they were clearly important to Real Madrid, too.
When I started looking a little closer at Real Madrid's soccer team, I began to suspect that perhaps they had a mathematician on the bench. A little analysis revealed that at the time of Beckham's move, all the Galácticos, the key players for Real Madrid, were playing in prime-number shirts: Carlos (the building block of the defense) wore number 3; Zidane (the heart of the midfield) was number 5; and Raúl and Ronaldo (the foundations of Real's strikers) sported numbers 7 and 11, respectively. So perhaps it was inevitable that Beckham also got a prime number, one that he has become very attached to. When he joined LA Galaxy, he insisted on taking his prime number with him in his attempt to woo the American public with the beautiful game.
This may sound totally irrational coming from a mathematician, someone who is meant to be a logical analytical thinker. However, I also play in a prime-number shirt for my soccer team, Recreativo Hackney, so I felt some connection with the man in 23. My Sunday League team isn't quite as big as Real Madrid and we didn't have a 23 shirt, so I chose 17—a rather nice prime, as we'll see later. But in our first season together, our team didn't do particularly well. We play in the London Super Sunday League Division 2, and that season we finished rock bottom. Fortunately, this is the lowest division in London, so the only way to go was up.
But how were we to improve our league standing? Maybe Real Madrid was on to something—was there some psychological advantage to be had from playing in a prime-number shirt? Perhaps too many of us were in nonprimes, like 8, 10, or 15. The next season, I persuaded the team to change our gear, and we all played in prime numbers: 2, 3, 5, 7 ... all the way up to 43. It transformed us. We got promoted to Division 1, where we quickly learned that primes last only for one season. We were relegated back down to Division 2, and we are now on the lookout for a new mathematical theory to boost our chances.
SHOULD REAL MADRID'S GOALKEEPER WEAR THE NUMBER 1 SHIRT?
If the key players for Real Madrid wear primes, then what shirt should the goalkeeper wear? Or, put mathematically, is 1 a prime? Well, yes and no. (This is just the sort of math question everyone loves—both answers are right.) Two hundred years ago, tables of prime numbers included 1 as the first prime. After all, it isn't divisible, since the only whole number that divides it is itself. But today, we say that 1 is not a prime because the most important thing about primes is that they are the building blocks of numbers. If I multiply a number by a prime, I get a new number. Although 1 is not divisible, if I multiply a number by 1, I get the number I started with, and on that basis, we exclude 1 from the list of primes and start at 2.
Clearly, Real Madrid wasn't the first to discover the potency of the primes. But which culture got there first—the ancient Greeks? The Chinese? The Egyptians? It turns out that mathematicians were beaten to the discovery of the primes by a strange little insect.
WHY DOES AN AMERICAN SPECIES OF CICADA LIKE THE PRIME 17?
In the forests of North America, there is a species of cicada with a very strange life cycle. For 17 years, these cicadas hide underground doing very little except sucking on the roots of the trees. Then in May of the seventeenth year, they emerge at the surface en masse to invade the forest—up to a million of them per acre.
The cicadas sing away to one another, trying to attract mates. Together, they make so much noise that local residents often move out for the duration of this invasion every 17 years. Bob Dylan was inspired to write his song "Day of the Locusts" when he heard the cacophony of cicadas that emerged in the forests around Princeton when he was collecting an honorary degree from the university in 1970.
After they've attracted a mate and become fertilized, the females each lay about six hundred eggs above ground. Then, after six weeks of partying, the cicadas all die and the forest goes quiet again for another 17 years. The next generation of eggs hatches in midsummer, and the nymphs drop to the forest floor before burrowing through the soil until they find a root to feed from. Then they wait another 17 years for the next great cicada party.
It's an absolutely extraordinary feat of biological engineering that these cicadas can count the passage of 17 years. It's very rare for any cicada to emerge a year early or a year too late. The annual cycle that most animals and plants work to is controlled by changing temperatures and the seasons. There is nothing that is obviously keeping track of the fact that the earth has gone around the sun 17 times and can then trigger the emergence of these cicadas.
For a mathematician, the most curious feature is the choice of number: 17, a prime number. Is it just a coincidence that these cicadas have chosen to spend a prime number of years hiding underground? It doesn't seem so. There are other species of cicada that stay underground for 13 years, and a few that prefer to stay there for 7 years—all prime numbers. Rather amazingly, if a 17-year cicada does appear too early, then it isn't out one year early, but generally four years early, apparently shifting to a 13-year cycle. There really does seem to be something about prime numbers that is helping these various species of cicada. But what is it?
While scientists aren't too sure, there is a mathematical theory that has emerged to explain the cicadas' addiction to primes. First, a few facts: A forest has, at most, one brood of cicadas, so the explanation isn't about sharing resources between different broods. In most years, a brood of prime- number cicadas emerges somewhere in the United States. However, 2009 and 2010 were cicada-free. In contrast, 2011 will see a massive brood of 13-year cicadas appearing in the southeastern United States. (Incidentally, 2011 is a prime, but I don't think the cicadas are that clever.)
The best theory to date for the cicadas' prime-number life cycle is the possible existence of a predator that also used to appear periodically in the forest, timing its arrival to coincide with the cicadas' and then feasting on the newly emerged insects. This is where natural selection kicks in, because cicadas that regulate their lives on a prime-number cycle are going to meet predators far less often than non-prime-number cicadas will.
For example, suppose that the predators appear every six years. Cicadas that appear every seven years will coincide with the predators only every 42 years. In contrast, cicadas that appear every eight years will coincide with the predators every 24 years; cicadas appearing every nine years will coincide even more frequently: every 18 years.
Across the forests of North America, there seems to have been real competition to find the biggest prime. The cicadas have been so successful that the predators have either starved or moved out, leaving the cicadas with their strange prime-number life cycle. But as we shall see, cicadas are not the only ones to have exploited the syncopated rhythm of the primes.
HOW ARE THE PRIMES 17 AND 29 THE KEY TO THE END OF TIME?
During the Second World War, the French composer Olivier Messiaen was incarcerated as a prisoner of war in Stalag VIII-A, where he discovered a clarinetist, a cellist, and a violinist among his fellow inmates. He decided to compose a quartet for these three musicians and himself on piano. The result was one of the great works of twentieth-century music: Quatuor pour la fin du temps (Quartet for the End of Time). It was first performed for inmates and prison officers inside Stalag VIII-A, with Messiaen playing a rickety upright piano they found in the camp.
In the first movement, called "Liturgie de Crystal," Messiaen wanted to create a sense of never-ending time, and the primes 17 and 29 turned out to be the key. While the violin and clarinet exchange themes representing birdsong, the cello and piano provide the rhythmic structure. In the piano part, there is a 17-note rhythmic sequence repeated over and over, and the chord sequence that is played on top of this rhythm consists of 29 chords. So as the 17-note rhythm starts for the second time, the chord sequence is just about two-thirds of the way through. The effect of the choice of prime numbers 17 and 29 is that the rhythmic and chordal sequences wouldn't repeat themselves until 17 × 29 notes through the piece.
It is this continually shifting music that creates the sense of timelessness that Messiaen was keen to establish—and he used the same trick as the cicadas with their predators. Think of the cicadas as the rhythm and the predators as the chords. The different primes, 17 and 29, keep the two out of sync so that the piece finishes before you ever hear the music repeat itself. Messiaen wasn't the only composer to have utilized prime numbers in music. Alban Berg also used a prime number as a signature in his music. Just like David Beckham, Berg sported the number 23—in fact, he was obsessed by it. For example, in his Lyric Suite, 23-bar sequences make up the structure of the piece. But embedded inside the piece is a representation of a love affair that Berg was having with a rich married woman. His lover was denoted by a ten-bar sequence that he entwined with his own signature 23, using the combination of mathematics and music to bring alive his affair.
Like Messiaen's use of primes in the Quartet for the End of Time, mathematics has recently been used to create a piece that although not timeless, nevertheless won't repeat itself for a thousand years. To mark the turn of the new millennium, Jem Finer, a founding member of The Pogues, decided to create a music installation in the East End of London that would repeat itself for the first time at the turn of the next millennium—3000. It's called, appropriately, Long player.
Finer started with a piece of music created with Tibetan singing bowls and gongs of different sizes. The original source music is 20 minutes and 20 seconds long, and by using some mathematics similar to the tricks employed by Messiaen, he expanded it into a piece that is a thousand years long. Six copies of the original source music are played simultaneously but at different speeds. In addition, every 20 seconds, each track is restarted a set distance from the original playback, but the amount by which each track is shifted is different. It is in the decision of how much to shift each track that the mathematics is used to guarantee that the tracks won't align perfectly again until a thousand years later.
It's not just musicians who are obsessed with prime numbers: these numbers seem to strike a chord with practitioners in many different fields of the arts. The author Mark Haddon only used prime-number chapters in his best-selling book, The Curious Incident of the Dog in the Night-Time. The narrator of the story is a boy named Christopher, who has Asperger's syndrome. Christopher likes the mathematical world because he can understand how it will behave—the logic of this world means there are no surprises. Human interactions, though, are full of the uncertainties and illogical twists that Christopher can't cope with. As Christopher explains, "I like prime numbers ... I think prime numbers are like life. They are very logical but you could never work out the rules, even if you spent all your lifetime thinking about them." (Continues...)
Excerpted from The Number Mysteries by Marcus Du Sautoy. Copyright © 2011 Marcus du Sautoy. Excerpted by permission of Palgrave Macmillan.
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