The Number Mysteries: A Mathematical Odyssey through Everyday Life

by Marcus du Sautoy

  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.

  Figure 1.2 The interaction over a hundred years between populations of cicadas with a seven-year life cycle and predators with a six-year life cycle.

  Figure 1.3 The interaction over a hundred years between populations of cicadas with a nine-year life cycle and predators with a six-year life cycle.

  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.

  Cicadas vs. Predators

  Download the PDF file for the cicada game from the Number Mysteries website (http://www.fifthestate.co.uk/numbermysteries/). Use the snakes-and-ladders board that can be downloaded from the same website. Cut out the predators and the two cicada families. Place the predators on the numbers in the 6 times table. Each player takes a family of cicadas. Take three standard six-sided dice. The roll of the dice will determine how often your family of cicadas appears. For example, if you roll an 8, then place cicadas on each number in the 8 times table. But if there is a predator already on a number, you can’t place a cicada—for example, you can’t place a cicada on 24 because it’s already occupied by a predator. The winner is the person with the most cicadas on the board. You can vary the game by changing the period of the predator from 6 to some other number.

  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.

  Figure 1.4 Messiaen’s “Liturgie de Crystal” from the Quatuor pour la fin du temps. The first vertical line indicates where the 17-note rhythm sequence ends. The second line indicates the end of the 29-note harmonic sequence. Property of Editions Durand, Paris. Reproduced by arrangement with G

  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, Longplayer.

  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.

  You can listen to Longplayer at http://longplayer.org or by using your smart-phone to scan this code.

  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.”

  Prime numbers have even had an outing in the movies. In the futuristic thriller Cube, seven characters are trapped in a maze of rooms, which resembles a complex Rubik’s Cube. Each room in the maze is cube-shaped with six doors leading to more rooms in the maze. The film begins with the characters waking up to find themselves inside this maze. They have no idea how they got there, but they have to find a way out. The problem is that some of the rooms are booby-trapped. The characters need to find some way of telling whether a room is safe before they enter it, for a whole array of horrific deaths await them, including being incinerated, getting covered in acid, and being cheese-wired into tiny cubes—as they discover when one of them is killed.

  One of the characters, Joan, is a mathematical whiz, and she suddenly sees that the numbers at the entrance to each room hold the key to revealing whether a trap lies ahead. It seems that if any of the numbers at the entrance are prime, then the room contains a trap. “You beautiful brain,” declares the leader of the group at this piece of mathematical deduction. It turns out that they also have to watch out for prime powers, but this proves to be beyond the clever Joan. Instead, they have to rely on one of their group who is an autistic savant, and he turns out to be the only one to make it out of the prime-number maze alive.

  As the cicadas discovered, knowing your math is the key to survival in this world. Teachers who are having trouble motivating their mathematics class might find some of the gory deaths in Cube to be a great piece of propaganda for getting their students to learn their primes.

  WHY DO SCIENCE FICTION WRITERS LIKE PRIMES?

  When science fiction writers want to get their aliens to communicate with Earth, they have a problem. Do they assume that their aliens are really clever and have picked up the local language, or that they’ve invented some clever Babel Fish–style translator that does the interpreting for them? Or do they just assume that everyone in the universe speaks English?

  One solution that a number of authors have gone for is that mathematics is the only truly universal language, and the first words that anyone should speak in this language are its building blocks—the primes. In Carl Sagan’s novel Contact, Ellie Arroway, who works for SETI (Search for Extraterrestrial Intelligence), picks up a signal that she realizes is not just background noise but a series of pulses. She guesses that they are binary representations of numbers. As she converts them into decimal numbers, she suddenly spots a pattern: 59, 61, 67, 71 . . . —all prime numbers. Sure enough, as the signal continues, it cycles through all the primes up to 907. This can’t be random, she concludes. Someone is saying hello.

  Many mathematicians believe that even if there is a different biology, a different chemistry, even a different physics on the other side of the universe, the mathematics will be the same. Anyone sitting on a planet orbiting Vega reading a math book about primes will still consider 59 and 61 to be prime numbers because, as the famous Cambridge mathematician G. H. Hardy put it, these numbers are prime “not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.”

  The primes may be numbers that are shared across the universe, but it is still interesting to wonder whether stories similar to those I’ve related are being told on other worlds. The way we have studied these numbers over the millennia has led us to discover important truths about them. And at each step on the way to discovering these truths, we can see the mark of a particular cultural perspective, the mathematical motifs of that period in history. Could other cultures across the universe have developed different perspectives, giving them access to theorems we have yet to discover?

  Carl Sagan wasn’t the first—and won’t be the last—to suggest using the primes as a way of communicating. Prime numbers have even been used by NASA in its attempts to make contact with extraterrestrial intelligence. In 1974, the Arecibo radio telescope in Puerto Rico broadcast a message toward the globular star cluster M13, chosen for its huge number of stars so as to increase the chance that the message might fall on intelligent ears.

  The message consisted of a series of 0s and 1s that could be arranged to form a black-and-white pixelated picture. The reconstructed image depicted the numbers from 1 to 10 in binary, a sketch of the structure of DNA, a representation of our solar system, and a picture of the Arecibo radio telescope itself. Considering that there were only 1,679 pixels, the picture is not very detailed. But the choice of 1,679 was deliberate because it contained the clue to setting up the pixels. 1,679 = 23 × 73, so there are only two ways to arrange the pixels in a rectangle to make up the picture. Arranging 23 rows of 73 columns produces a jumbled mess, but arrange them the other way—as 73 rows of 23 columns—and you get the result shown in Figure 1.5. The star cluster M13 is 25,000 light years away, so we’re still waiting for a reply. Don’t expect a response for another 50,000 years!

  Figure 1.5 The message broadcast by the Arecibo radio telescope toward the star cluster M13.

  Although the primes are universal, the way we write them has varied greatly throughout the history of mathematics, and is very culture-specific—as our whistle-stop tour of the planet will now illustrate.

  Which prime is this?

  Figure 1.6

  Some of the first mathematics in history was done in ancient Egypt, and this is how they wrote the number 200,201. As early as 6000 BC, people were abandoning nomadic life to settle along the Nile River. As Egyptian society became more sophisticated, the need grew for numbers to record taxes, measure land, and construct pyramids. Just as for their language, the Egyptians used hieroglyphs to write numbers. They had already developed a number system based on powers of 10, like the decimal system we use today. (The choice comes not from any special mathematical significance of the number, but from the anatomical fact that we have ten fingers.) But they had yet to invent the place-value system, which is a way of writing numbers so that the position of each digit corresponds to the power of 10 that the digit is counting. For example, the 2s in 222 all have different values according to their different positions. Instead, the Egyptians needed to create new symbols for each new power of 10:

  Figure 1.7 Ancient Egyptian symbols for powers of 10. 10 is a stylized heel bone, 100 a coil of rope, and 1,000 a lotus plant.

  200,201 can be written quite economically in this way, but just try writing the prime 9,999,991 in hieroglyphs: you would need 55 symbols. Although the Egyptians did not realize the importance of the primes, they did develop some sophisticated math, including—not surprisingly—the formula for the volume of a pyramid and a concept of fractions. But their notation for numbers was not very sophisticated, unlike the one used by their neighbors, the Babylonians.

  Which prime is this?

  Figure 1.8

  This is how the ancient Babylonians wrote the number 71. Like the Egyptian empire, the Babylonian empire was focused around a major river: the Euphrates. From 1800 BC, the Babylonians controlled much of modern Iraq, Iran, and Syria. To expand and run their empire, they became masters of managing and manipulating numbers. Records were kept on clay tablets, and scribes would use a wooden stick or stylus to make marks in the wet clay, which would then be dried. The tip of the stylus was wedge-shaped, or cuneiform—the name by which the Babylonian script is now known.

  Around 2000 BC, the Babylonians became one of the first cultures to use the idea of a place-value number
system. But instead of using powers of 10 like the Egyptians, the Babylonians developed a number system that worked in base 60. They had different combinations of symbols for all the numbers from 1 to 59. When they reached 60, they started a new “60s” column to the left and recorded one lot of 60, in the same way that in the decimal system, we place a 1 in the “10s” column when the units column passes 9. So the prime number shown in Figure 1.8 consists of one lot of 60 together with the symbol for 11, making 71. The symbols for the numbers up to 59 do have some hidden appeal to the decimal system because the numbers from 1 to 9 are represented by horizontal lines, but then 10 is represented by the following symbol:

  Figure 1.9

  The choice of base 60 is much more mathematically justified than the decimal system. It is a highly divisible number, which makes it very powerful for doing calculations. For example, if I have 60 beans, I can divide them up in a multitude of different ways:

  60 = 30 × 2 = 20 × 3 = 15 × 4 = 12 × 5 = 10 × 6

  Figure 1.10 The different ways of dividing up 60 beans.

  The Babylonians came close to discovering a very important number in mathematics: 0. If you wanted to write the prime number 3,607 in cuneiform, you had a problem. This is one lot of 3,600—or 60 squared—and 7 units, but if I write that down, it could easily look like one lot of 60 and 7 units—still a prime, but not the prime I want. To get around this, the Babylonians introduced a little symbol to denote that there were no 60s being counted in the 60s column. So 3,607 would be written as