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. The number of pairs of rabbits in successive months is given by the following sequence:
FIGURE 1.22 The Fibonacci numbers are the key to calculating the population growth of rabbits.
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. Trillium has three, a pansy has five, a delphinium has eight, marigolds have 13, chicory has 21, pyrethrum 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 round exceptions. (I don’t want to be inundated with letters from irate gardeners, so I’ll concede that there are a few exceptions which 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 3 segments. Cut open an apple with a slice halfway between the stalk and the base, and you’ll see a 5-pointed star. Try the same with a Sharon fruit, and you’ll get an 8-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.
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.
FIGURE 1.23 How to build a shell using Fibonacci numbers.
In the eighth century AD the Indian writer Virahanka took up the challenge to determine 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 got 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 pth Fibonacci number, where p is a prime number, is itself prime. For example, 11 is prime, and the 11th 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 19th Fibonacci number is 4,181, and although 19 is prime, 4,181 is not: it equals 37×113. No mathematician has yet proved whether infinitely many 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 1 grain of rice to be placed on the first square of the chessboard, 2 on the second, 4 on the third, 8 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 got to the 16th square, he was already needing another kilogram of rice. By the 20th square, his servants had to bring in a wheelbarrow full. He never reached the 64th and last square on the board. By that point the total number of grains of rice on the board would have been a staggering
18,446,744,073,709,551,615
If we tried to repeat the feat at the heart of London, the pile of rice on the 64th square would stretch to the boundaries of the M25 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.
FIGURE 1.24 Repeated doubling makes numbers grow very quickly.
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 maths can make you rich.
But what has all this rice got to do with finding big prime numbers? Ever since the Greeks had proved that the primes go on for ever, mathematicians had been on the look-out for clever formulas that might generate bigger and bigger primes. One of the best of these
