The Number Mysteries: A Mathematical Odyssey through Everyday Life

by Marcus du Sautoy

  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 plow through 59 nonprimes to get to the pair 102,761 and 102,763. 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 an infinite number of primes. But are we going to keep coming across twin primes? As yet, nobody has come up with a clever proof like Euclid’s to show whether there are an infinite number of these twin primes.

  At one stage, it seemed that human 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. Studying 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, October 23, 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; 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 and divide the result by p. If the remainder isn’t 2, then p can’t be prime. Some people have speculated that, given the twins’ aptitude for identifying days of the week—which depends on a similar technique of looking at remainders upon division by 7—they may well have been using this test to find primes.

  At first, mathematicians thought that if 2p does have a remainder of 2 upon division by p, then p must be prime. But it turns out that this test does not guarantee primality. 341 = 31 × 11 is not prime, yet 2341 has a remainder of 2 upon division by 341. This example was not discovered until 1819, and it is possible that the twins might have been aware of a more sophisticated test that would wheedle out 341. Fermat showed that the test can be extended past powers of 2 by proving that if p is prime, then for any number n less than p, np always has a remainder of n when divided by the prime p. So if you find any number n for which this fails, you can throw out p as a prime impostor.

  For example, 3341 doesn’t have a remainder of 3 upon division by 341—it has a remainder of 168. The twins couldn’t possibly have been checking through all numbers less than their candidate prime: there would be too many tests for them to run through. However, the great Hungarian prime-number wizard, Paul Erdös, estimated (though he couldn’t prove it rigorously) that to test whether a number less than 10150 is prime, passing Fermat’s test just once means that the chances of the number not being prime are as low as 1 in 1043. So for the twins, probably one test was enough to give them the buzz of prime discovery.

  PRIME-NUMBER HOPSCOTCH

  This is a game for two players in which knowing your twin primes can give you an edge.

  Write down the numbers from 1 to 100, or use the snakes-and-ladders board, which you can download from the Number Mysteries website. The first player takes a counter and places it on a prime number, which is, at most, five steps away from square 1. The second player takes the counter and moves it to a bigger prime that is, at most, five squares ahead of where the first player placed it. The first player follows suit, moving the counter to an even higher prime number that again is, at most, five squares ahead. The loser is the first player unable to move the counter according to the rules. The rules are the following: (1) the counter can’t be moved farther than five squares ahead, (2) it must always be moved to a prime, and (3) it can’t be moved backward or left where it is.

  Figure 1.21 An example of a prime-number hopscotch game in which the maximum move is five steps.

  Figure 1.21 shows a typical scenario. The first player has lost the game because the counter is at 23 and there are no primes in the five numbers ahead of 23. Could the first player have made a better opening move? If you look carefully, you’ll see that once you’ve passed 5, there really aren’t many choices. Whoever moves the counter to 5 is going to win because that player will, at a later turn, be able to move the counter from 19 to 23, leaving his or her opponent with no prime to move to. So the opening move is vital.

  But what if we change the game a little? Let’s say that you are allowed to move the counter to a prime that is, at most, seven steps ahead. Players can now jump a little farther. In particular, they can get past 23 because 29 is six steps ahead and within reach. Does your opening move matter this time? Where will the game end? If you play the game, you’ll find that this time you have many more choices along the way, especially when there is a pair of twin primes.

  At first glance, with so many choices, it looks like your first move is irrelevant. But look again. You lose if you find yourself on 89, because the next prime after 89 is 97—eight steps ahead. If you trace your way back through the primes, you’ll find that being on 67 is crucial because here you get to choose which of the twin primes 71 and 73 you place the counter on. One is a winning choice; the other will make you lose the game because every move from that point on is forced on you. Whoever is on 67 can win the game, and it seems that 89 is not so important. So how can you make sure you get there?

  If you continue tracing your way back through the game, you’ll find that there’s a crucial decision to be made for anyone on the prime 37. From there, you can reach my daughters’ twin primes, 41 and 43. Move to 41, and you can guarantee winning the game. So now it looks as if the game is decided by whoever can get to the prime 37. Continuing to wind the game back in this way reveals that there is indeed a winning opening move. Put the counter on 5, and from there you can guarantee that you get all the crucial decisions that ensure you get to move the counter to 89 and win the game—because then your opponent can’t move.

  What if we continue to make the maximum permitted jumps even bigger: can we always be sure that the game will end? What if we allow each player to move a maximum of 99 steps—can we be sure that the game won’t just go on forever because you can always jump to another prime within 99 of the last one? After all, we know that there are an infinite number of primes, so perhaps at some point, you can simply jump from one prime to the next.

  It is actually possible to prove that the game does always end. However far you set the maximum jump, there will always be a stretch of numbers greater than the maximum jump containing no primes—it’s there that the game will end. Let’s look at how to find 99 consecu
tive numbers, none of which are prime. Take the number 100 × 99 × 98 × 97 × . . . × 3 × 2 × 1. This number is known as 100 factorial and is written as 100! We’re going to use an important fact about this number: if you take any number between 1 and 100, then 100! is divisible by this number. Look at this sequence of consecutive numbers:

  100! + 2, 100! + 3, 100! + 4, . . . , 100! + 98, 100! + 99, 100! + 100

  100! + 2 is not prime because it is divisible by 2. Similarly, 100! + 3 is not prime because it is divisible by 3. (100! is divisible by 3, so if we add 3, it’s still divisible by 3.) In fact, none of these numbers is prime. Take 100! + 53, which is not prime because 100! is divisible by 53, and if we add 53, the result is still divisible by 53. Here are 99 consecutive numbers, none of which is prime. The reason we started at 100! + 2 and not 100! + 1 is that with this simple method, we can deduce only that 100! + 1 is divisible by 1, and that won’t help us to tell whether it’s prime. (In fact, it isn’t.)

  So we know for certain that if we set the maximum jumps to 99, our prime-number hopscotch game will end at some point. But 100! is a ridiculously large number. The game would actually finish way before this point: the first place where a prime is followed by 99 nonprimes is 396,733.

  Playing this game certainly reveals the erratic way in which the primes seem to be scattered through the universe of numbers. At first glance, there’s no way of knowing where to find the next prime. But if we can’t find a clever device for navigating from one prime to the next, can we at least come up with some clever formulas to produce primes?

  This website has information about where the hopscotch game will end for larger and larger jumps: www.trnicely.net/gaps/gaplist.html#MainTable. You can use your smartphone to scan this code.

  COULD RABBITS AND SUNFLOWERS BE USED TO FIND PRIMES?

  Count the number of petals on a sunflower. Often, there are 89—a prime number. The number of pairs of rabbits after 11 generations is also 89. Have rabbits and flowers discovered some secret formula for finding primes? Not exactly. They like 89 not because it is prime, but because it is one of nature’s other favorite numbers: the Fibonacci numbers. The Italian mathematician Fibonacci of Pisa discovered this important sequence of numbers in 1202 when he was trying to understand the way rabbits multiply (in the biological, rather than the mathematical, sense).

  Fibonacci started by imagining a pair of baby rabbits—one male, one female. Call this starting point month 1. By month 2, these rabbits have matured into an adult pair, which can breed and produce in month 3 a new pair of baby rabbits. (For the purposes of this thought experiment, all litters consist of one male and one female.) In month 4, the first adult pair produce another pair of baby rabbits. Their first pair of baby rabbits has now reached adulthood, so there are now two pairs of adult rabbits and a pair of baby rabbits. In month 5, the two pairs of adult rabbits each produce a pair of baby rabbits. The baby rabbits from month 4 become adults. So by month 5, there are three pairs of adult rabbits and two pairs of baby rabbits, making five pairs of rabbits in total.

  Figure 1.22 The Fibonacci numbers are the key to calculating the population growth of rabbits.

  The number of pairs of rabbits in successive months is given by the following sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . Keeping track of all these multiplying rabbits was quite a headache until Fibonacci spotted an easy way to work out the numbers. To get the next number in the sequence, you just add the two previous numbers. The bigger of the two is, of course, the number of pairs of rabbits up to that point. They all survive to the next month, and the smaller of the two is the number of adult pairs. These adult pairs each produce an extra pair of baby rabbits, so the number of rabbits in the next month is the sum of the numbers in the two previous generations.

  Some readers might recognize this sequence from Dan Brown’s novel The Da Vinci Code. They are in fact the first code that the hero has to crack on his way to the Holy Grail.

  It isn’t only rabbits and Dan Brown who like these numbers. The number of petals on a flower is often a Fibonacci number. Trilliums have three, pansies have five, delphiniums have eight, marigolds have 13, chicories have 21, pyrethrums have 34, and sunflowers often have 55 or even 89 petals. Some plants have flowers with twice a Fibonacci number of petals. These are plants, like some lilies, that are made up of two copies of a flower. And if your flower doesn’t have a Fibonacci number of petals, then that’s because a petal has fallen off . . . which is how mathematicians get around exceptions. (I don’t want to be inundated with letters from irate gardeners, so I’ll concede that there are a few exceptions that aren’t just examples of wilting flowers. For example, the starflower often has seven petals. Biology is never as perfect as mathematics.)

  As well as in flowers, you can find the Fibonacci numbers running up and down pine cones and pineapples. Slice across a banana, and you’ll find that it’s divided into three segments. Cut open an apple with a slice halfway between the stalk and the base, and you’ll see a five-pointed star. Try the same with a Sharon fruit, and you’ll get an eight-pointed star. Whether it’s populations of rabbits or the structures of sunflowers or fruit, the Fibonacci numbers seem to crop up whenever there is growth happening.

  The way shells evolve is also closely connected to these numbers. A baby snail starts off with a tiny shell, effectively a little one-by-one square house. As it outgrows its shell, it adds another room to the house and repeats the process as it continues to grow. Since it doesn’t have much to go on, it simply adds a room whose dimensions are based on those of the two previous rooms, just as Fibonacci numbers are the sum of the previous two numbers. The result of this growth is a simple but beautiful spiral.

  Figure 1.23 How to build a shell using Fibonacci numbers.

  Actually, these numbers shouldn’t be named after Fibonacci at all, because he was not the first to stumble across them. In fact, they weren’t discovered by mathematicians at all, but by poets and musicians in medieval India. Indian poets and musicians were keen to explore all the possible rhythmic structures you can generate by using combinations of short and long rhythmic units. If a long sound is twice the length of a short sound, then how many different patterns are there with a set number of beats? For example, with eight beats you could do four long sounds or eight short ones. But there are lots of combinations between these two extremes.

  In the eighth century AD, the Indian writer Virahanka took on the challenge of determining exactly how many different rhythms are possible. He discovered that as the number of beats goes up, the number of possible rhythmic patterns is given by the following sequence: 1, 2, 3, 5, 8, 13, 21, . . . He realized, just as Fibonacci did, that to get the next number in the sequence, you simply add together the two previous numbers. So if you want to know how many possible rhythms there are with eight beats, you go to the eighth number in the sequence, which is obtained by adding 13 and 21 to arrive at 34 different rhythmic patterns.

  Perhaps it’s easier to understand the mathematics behind these rhythms than to try to follow the increasing population of Fibonacci’s rabbits. For example, to get the number of rhythms with eight beats, you take the rhythms with six beats and add a long sound or take the rhythms with seven beats and add a short sound.

  There is an intriguing connection between the Fibonacci sequence and the protagonists of this chapter—the primes. Look at the first few Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . Every p th Fibonacci number—where p is a prime number—is itself prime. For example, 11 is prime, and the eleventh Fibonacci number is 89—a prime. If this always worked, it would be a great way to generate bigger and bigger primes. Unfortunately, it doesn’t. The nineteenth Fibonacci number is 4,181, and although 19 is prime, 4,181 is not: it equals 37 × 113. No mathematician has yet proved whether an infinite number of Fibonacci numbers are prime numbers. This is another of the many unsolved prime-number mysteries in mathematics.

  HOW CAN YOU USE RICE AND A CHESSBOARD TO FIND PRIMES?

/>   Legend has it that chess was invented in India by a mathematician. The king was so grateful to the mathematician that he asked him to name any prize as a reward. The inventor thought for a minute, then asked for one grain of rice to be placed on the first square of the chessboard, two on the second, four on the third, eight on the fourth, and so on, so that each square got twice as many grains of rice as were on the previous square.

  The king readily agreed, astonished that the mathematician wanted so little—but he was in for a shock. When he began to place the rice on the board, the first few grains could hardly be seen. But by the time he’d gotten to the sixteenth square, he already needed another kilogram of rice. By the twentieth square, his servants had to bring in a wheelbarrow full. He never reached the sixty-fourth and last square on the board. By that point, the total number of grains of rice would have been a staggering 18,446,744,073,709,551,615.

  Figure 1.24 Repeated doubling makes numbers grow very quickly.

  If we tried to repeat the feat at the heart of London, the pile of rice on the sixty-fourth square would stretch to a distance of 15 miles from the center and would be so high that it would cover all the buildings. In fact, there would be more rice in this pile than has been produced across the globe in the last millennium.

  Not surprisingly, the king of India failed to give the mathematician the prize he had been promised and was forced into parting with half his fortune instead. That’s one way math can make you rich.